What is a word equation in math

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.^[2][3] The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.^[4]

Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables.^[5][6]

An equation is written as two expressions, connected by an equals sign («=»).^[2] The expressions on the two sides of the equals sign are called the «left-hand side» and «right-hand side» of the equation. Very often the right-hand side of an equation is assumed to be zero. This does not reduce the generality, as this can be realized by subtracting the right-hand side from both sides.

The most common type of equation is a polynomial equation (commonly called also an algebraic equation) in which the two sides are polynomials.
The sides of a polynomial equation contain one or more terms. For example, the equation

has left-hand side , which has four terms, and right-hand side , consisting of just one term. The names of the variables suggest that x and y are unknowns, and that A, B, and C are parameters, but this is normally fixed by the context (in some contexts, y may be a parameter, or A, B, and C may be ordinary variables).

An equation is analogous to a scale into which weights are placed. When equal weights of something (e.g., grain) are placed into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. More generally, an equation remains in balance if the same operation is performed on its both sides.

In Cartesian geometry, equations are used to describe geometric figures. As the equations that are considered, such as implicit equations or parametric equations, have infinitely many solutions, the objective is now different: instead of giving the solutions explicitly or counting them, which is impossible, one uses equations for studying properties of figures. This is the starting idea of algebraic geometry, an important area of mathematics.

Algebra studies two main families of equations: polynomial equations and, among them, the special case of linear equations. When there is only one variable, polynomial equations have the form P(x) = 0, where P is a polynomial, and linear equations have the form ax + b = 0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis. Algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.

Differential equations are equations that involve one or more functions and their derivatives. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in areas such as physics, chemistry, biology, and economics.

The «=» symbol, which appears in every equation, was invented in 1557 by Robert Recorde, who considered that nothing could be more equal than parallel straight lines with the same length.^[1]

Introduction[edit]

Analogous illustration[edit]

An equation is analogous to a weighing scale, balance, or seesaw.

Each side of the equation corresponds to one side of the balance. Different quantities can be placed on each side: if the weights on the two sides are equal, the scale balances, and in analogy, the equality that represents the balance is also balanced (if not, then the lack of balance corresponds to an inequality represented by an inequation).

In the illustration, x, y and z are all different quantities (in this case real numbers) represented as circular weights, and each of x, y, and z has a different weight. Addition corresponds to adding weight, while subtraction corresponds to removing weight from what is already there. When equality holds, the total weight on each side is the same.

Parameters and unknowns[edit]

Equations often contain terms other than the unknowns. These other terms, which are assumed to be known, are usually called constants, coefficients or parameters.

An example of an equation involving x and y as unknowns and the parameter R is

When R is chosen to have the value of 2 (R = 2), this equation would be recognized in Cartesian coordinates as the equation for the circle of radius of 2 around the origin. Hence, the equation with R unspecified is the general equation for the circle.

Usually, the unknowns are denoted by letters at the end of the alphabet, x, y, z, w, …, while coefficients (parameters) are denoted by letters at the beginning, a, b, c, d, … . For example, the general quadratic equation is usually written ax² + bx + c = 0.

The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called solving the equation. Such expressions of the solutions in terms of the parameters are also called solutions.

A system of equations is a set of simultaneous equations, usually in several unknowns for which the common solutions are sought. Thus, a solution to the system is a set of values for each of the unknowns, which together form a solution to each equation in the system. For example, the system

has the unique solution x = −1, y = 1.

Identities[edit]

An identity is an equation that is true for all possible values of the variable(s) it contains. Many identities are known in algebra and calculus. In the process of solving an equation, an identity is often used to simplify an equation, making it more easily solvable.

In algebra, an example of an identity is the difference of two squares:

which is true for all x and y.

Trigonometry is an area where many identities exist; these are useful in manipulating or solving trigonometric equations. Two of many that involve the sine and cosine functions are:

and

which are both true for all values of θ.

For example, to solve for the value of θ that satisfies the equation:

where θ is limited to between 0 and 45 degrees, one may use the above identity for the product to give:

yielding the following solution for θ:

Since the sine function is a periodic function, there are infinitely many solutions if there are no restrictions on θ. In this example, restricting θ to be between 0 and 45 degrees would restrict the solution to only one number.

Properties[edit]

Two equations or two systems of equations are equivalent, if they have the same set of solutions. The following operations transform an equation or a system of equations into an equivalent one – provided that the operations are meaningful for the expressions they are applied to:

• Adding or subtracting the same quantity to both sides of an equation. This shows that every equation is equivalent to an equation in which the right-hand side is zero.
• Multiplying or dividing both sides of an equation by a non-zero quantity.
• Applying an identity to transform one side of the equation. For example, expanding a product or factoring a sum.
• For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity.

If some function is applied to both sides of an equation, the resulting equation has the solutions of the initial equation among its solutions, but may have further solutions called extraneous solutions. For example, the equation has the solution Raising both sides to the exponent of 2 (which means applying the function to both sides of the equation) changes the equation to , which not only has the previous solution but also introduces the extraneous solution, Moreover, if the function is not defined at some values (such as 1/x, which is not defined for x = 0), solutions existing at those values may be lost. Thus, caution must be exercised when applying such a transformation to an equation.

The above transformations are the basis of most elementary methods for equation solving, as well as some less elementary ones, like Gaussian elimination.

Algebra[edit]

Polynomial equations[edit]

In general, an algebraic equation or polynomial equation is an equation of the form

, or

^[a]

where P and Q are polynomials with coefficients in some field (e.g., rational numbers, real numbers, complex numbers). An algebraic equation is univariate if it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called multivariate (multiple variables, x, y, z, etc.).

For example,

is a univariate algebraic (polynomial) equation with integer coefficients and

is a multivariate polynomial equation over the rational numbers.

Some polynomial equations with rational coefficients have a solution that is an algebraic expression, with a finite number of operations involving just those coefficients (i.e., can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but equations of degree five or more cannot always be solved in this way, as the Abel–Ruffini theorem demonstrates.

A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root finding of polynomials) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).

Systems of linear equations[edit]

A system of linear equations (or linear system) is a collection of linear equations involving one or more variables.^[b] For example,

is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by

since it makes all three equations valid. The word «system» indicates that the equations are to be considered collectively, rather than individually.

In mathematics, the theory of linear systems is a fundamental part of linear algebra, a subject which is used in many parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in physics, engineering, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.

Geometry[edit]

Analytic geometry[edit]

In Euclidean geometry, it is possible to associate a set of coordinates to each point in space, for example by an orthogonal grid. This method allows one to characterize geometric figures by equations. A plane in three-dimensional space can be expressed as the solution set of an equation of the form , where and are real numbers and are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid. The values are the coordinates of a vector perpendicular to the plane defined by the equation. A line is expressed as the intersection of two planes, that is as the solution set of a single linear equation with values in or as the solution set of two linear equations with values in

A conic section is the intersection of a cone with equation and a plane. In other words, in space, all conics are defined as the solution set of an equation of a plane and of the equation of a cone just given. This formalism allows one to determine the positions and the properties of the focuses of a conic.

The use of equations allows one to call on a large area of mathematics to solve geometric questions. The Cartesian coordinate system transforms a geometric problem into an analysis problem, once the figures are transformed into equations; thus the name analytic geometry. This point of view, outlined by Descartes, enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.

Currently, analytic geometry designates an active branch of mathematics. Although it still uses equations to characterize figures, it also uses other sophisticated techniques such as functional analysis and linear algebra.

Cartesian equations[edit]

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, that are marked using the same unit of length.

One can use the same principle to specify the position of any point in three-dimensional space by the use of three Cartesian coordinates, which are the signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines).

The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 in a plane, centered on a particular point called the origin, may be described as the set of all points whose coordinates x and y satisfy the equation x² + y² = 4.

Parametric equations[edit]

A parametric equation for a curve expresses the coordinates of the points of the curve as functions of a variable, called a parameter.^[7][8] For example,

are parametric equations for the unit circle, where t is the parameter. Together, these equations are called a parametric representation of the curve.

The notion of parametric equation has been generalized to surfaces, manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.).

Number theory[edit]

Diophantine equations[edit]

A Diophantine equation is a polynomial equation in two or more unknowns for which only the integer solutions are sought (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An example of linear Diophantine equation is ax + by = c where a, b, and c are constants. An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns.

Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it.

The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.

Algebraic and transcendental numbers[edit]

An algebraic number is a number that is a solution of a non-zero polynomial equation in one variable with rational coefficients (or equivalently — by clearing denominators — with integer coefficients). Numbers such as π that are not algebraic are said to be transcendental. Almost all real and complex numbers are transcendental.

Algebraic geometry[edit]

Algebraic geometry is a branch of mathematics, classically studying solutions of polynomial equations. Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.

Differential equations[edit]

A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including physics, engineering, economics, and biology.

In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

Ordinary differential equations[edit]

An ordinary differential equation or ODE is an equation containing a function of one independent variable and its derivatives. The term «ordinary» is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.

Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.

Partial differential equations[edit]

A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model.

PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.

Types of equations[edit]

Equations can be classified according to the types of operations and quantities involved. Important types include:

• An algebraic equation or polynomial equation is an equation in which both sides are polynomials (see also system of polynomial equations). These are further classified by degree:
  • linear equation for degree one
  • quadratic equation for degree two
  • cubic equation for degree three
  • quartic equation for degree four
  • quintic equation for degree five
  • sextic equation for degree six
  • septic equation for degree seven
  • octic equation for degree eight
• A Diophantine equation is an equation where the unknowns are required to be integers
• A transcendental equation is an equation involving a transcendental function of its unknowns
• A parametric equation is an equation in which the solutions for the variables are expressed as functions of some other variables, called parameters appearing in the equations
• A functional equation is an equation in which the unknowns are functions rather than simple quantities
• Equations involving derivatives, integrals and finite differences:
  • A differential equation is a functional equation involving derivatives of the unknown functions, where the function and its derivatives are evaluated at the same point, such as . Differential equations are subdivided into ordinary differential equations for functions of a single variable and partial differential equations for functions of multiple variables
  • An integral equation is a functional equation involving the antiderivatives of the unknown functions. For functions of one variable, such an equation differs from a differential equation primarily through a change of variable substituting the function by its derivative, however this is not the case when the integral is taken over an open surface
  • An integro-differential equation is a functional equation involving both the derivatives and the antiderivatives of the unknown functions. For functions of one variable, such an equation differs from integral and differential equations through a similar change of variable.
  • A functional differential equation of delay differential equation is a function equation involving derivatives of the unknown functions, evaluated at multiple points, such as
  • A difference equation is an equation where the unknown is a function f that occurs in the equation through f(x), f(x−1), …, f(x−k), for some whole integer k called the order of the equation. If x is restricted to be an integer, a difference equation is the same as a recurrence relation
  • A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process

See also[edit]

Notes[edit]

References[edit]

External links[edit]

• Winplot: General Purpose plotter that can draw and animate 2D and 3D mathematical equations.
• Equation plotter: A web page for producing and downloading pdf or postscript plots of the solution sets to equations and inequations in two variables (x and y).

An equation is a mathematical statement with an ‘equal to’ symbol between two expressions that have equal values. For example, 3x + 5 = 15. There are different types of equations like linear, quadratic, cubic, etc. Let us learn more about equations in math in this article.

1.	What are Equations?
2.	Parts of an Equation
3.	How to Solve an Equation?
4.	Types of Equations
5.	Equation vs Expression
6.	FAQs on Equations

What are Equations?

Equations are mathematical statements containing two algebraic expressions on both sides of an ‘equal to (=)’ sign. It shows the relationship of equality between the expression written on the left side with the expression written on the right side. In every equation in math, we have, L.H.S = R.H.S (left hand side = right hand side). Equations can be solved to find the value of an unknown variable representing an unknown quantity. If there is no ‘equal to’ symbol in the statement, it means it is not an equation. It will be considered as an expression. You will learn the difference between equation and expression in the later section of this article.

Look at the following examples. These will give you an idea of the meaning of an equation in math.

	Equations	Is it an equation?
1.	y = 8x — 9	Yes
2.	y + x2 — 7	No, because there is no ‘equal to’ symbol.
3.	7 + 2 = 10 — 1	Yes

Now, let us move forward and learn about parts of an equation in math.

Parts of an Equation

There are different parts of an equation which include coefficients, variables, operators, constants, terms, expressions, and an equal to sign. When we write an equation, it is mandatory to have an «=» sign, and terms on both sides. Both sides should be equal to each other. An equation doesn’t need to have multiple terms on either of the sides, having variables, and operators. An equation can be formed without these as well, for example, 5 + 10 = 15. This is an arithmetic equation with no variables. As opposed to this, an equation with variables is an algebraic equation. Look at the image below to understand the parts of an equation.

How to Solve an Equation?

An equation is like a weighing balance with equal weights on both sides. If we add or subtract the same number from both sides of an equation, it still holds. Similarly, if we multiply or divide the same number into both sides of an equation, it still holds. Consider the equation of a line, 3x − 2 = 4. We will perform mathematical operations on the LHS and the RHS so that the balance is not disturbed. Let’s add 2 on both sides to reduce the LHS to 3x. This will not disturb the balance. The new LHS is 3x − 2 + 2 = 3x and the new RHS is 4 + 2 = 6. So, the equation becomes 3x = 6. Now, let’s divide both sides by 3 to reduce the LHS to x. Thus, the solution of the given equation of a line is x = 2.

The steps to solve a basic equation with one variable (linear) are given below:

• Step 1: Bring all the terms with variables on one side and all the constants on the other side of the equation by applying arithmetic operations on both sides.
• Step 2: Combine all like terms (terms containing the same variable with the same exponent) by adding/subtracting them.
• Step 3: Simplify it and get the answer.

Let us take one more example of a basic equation: 3x — 20 = 7. To bring all the constants on RHS, we have to add 20 to both sides. This implies, 3x — 20 + 20 = 7 + 20, which can be simplified as 3x = 27. Now, divide both sides by 3. This will get you x = 9 which is the required solution of the equation.

Types of Equations

Based on the degree, equations can be classified into three types. Following are the three types of equations in math:

• Linear Equations
• Quadratic Equations
• Cubic Equations

Linear Equation

Equations with 1 as the degree are known as linear equations in math. In such equations, 1 is the highest exponent of terms. These can be further classified into linear equations in one variable, two-variable linear equations, with three variables, etc. The standard form of a linear equation with variables X and Y are aX + bY — c = 0, where a and b are the coefficients of X and Y respectively and c is the constant.

Quadratic Equation

Equations with degree 2 are known as quadratic equations. The standard form of a quadratic equation with variable x is ax² + bx + c = 0, where a ≠ 0. These equations can be solved by splitting the middle term, completing the square, or by the discriminant method.

Cubic Equations

Equations with degree 3 are known as cubic equations. Here, 3 is the highest exponent of at least one of the terms. The standard form of a cubic equation with variable x is ax³ + bx² + cx + d = 0, where a ≠ 0.

Equation vs Expression

Expressions and equations in math are used simultaneously in algebra, but there is a major difference between the two terms. When 2x + 4 is an expression, 2x + 4 = 0 is considered an equation. Let us understand the basic difference between equation and expression through the table given below:

Equation	Expression
When two expressions are equal in value and written together with an ‘equal to’ sign in between, it is known as an equation in math.	It is a mathematical statement having at least one term or multiple terms connected through operators in between.
It has an equal to «=» sign.	An expression does not contain an equal to «=» sign.
It can be solved to find the value of the unknown quantity.	It can be simplified to the lowest form.
Example: x — 8 = 16, 6y = 33, 3z — 7y = 9, etc.	Example: x — 8, 6y, 3z — 7y — 9, etc.

Important Notes on Equations in Math:

• The values of the variable that makes an equation true are called the solution or root of the equation.
• The solution of an equation is unaffected if the same number is added, subtracted, multiplied, or divided into both sides of the equation.
• The graph of a linear equation in one or two variables is a straight line.
• The curve of the quadratic equation is in the form of a parabola.

☛ Related Topics:

Check these interesting articles related to the concept of equations in math.

• System of Equations
• Simple Equations and Its Applications
• Solve for x

FAQs on Equation

What is an Equation in Math?

An equation in math is an equality relationship between two expressions written on both sides of the equal to sign. For example, 3y = 16 is an equation.

What is a Linear Equation?

A linear equation is an equation with degree 1. It means the highest exponent of any term could be 1. An example of a linear equation in math is x + y = 24.

What is a Quadratic Equation?

A quadratic equation is an equation with degree 2. It can have any number of variables but the highest power of terms could be only 2. The standard form of a quadratic equation with variable y is ay² + by + c = 0, where a ≠ 0.

How Equations are Used in Real Life?

In real life, there are many situations in which equations can be used. Whenever an unknown quantity has to be found, an equation can be formed and solved. For example, if the cost of 1 pencil is $1.2, and the total money spent by you on pencils is $9.6, the number of pencils bought can be found by forming an equation based on the given information. Let the number of pencils bought be x. Then, the equation will be 1.2x = 9.6, which can be solved as x = 8.

How to Solve Quadratic Equations?

Quadratic equations in one variable can be solved using the following methods:

• Factorization method
• Completing the square method
• Discriminant method

What are the 3 Types of Equation?

Based on the degree, equations can be classified into the following three types:

• Linear equation
• Quadratic equation
• Cubic equation

What Equation Has No Solution?

Equations of two parallel lines have no solutions as they do not intersect at any point. To identify the equations of parallel lines, we have to compare the coefficients of both the variables in the given two linear equations in two variables. If the ratio of coefficients is the same and unequal to the ratio of constants, it means those equations have no solution. For two equations ax + by + c = 0 and px + qy + r = 0, it will have no solution when a/p = b/q ≠ c/r.

What is the Equation of a Circle?

The equation of a circle with radius r and center (x₁, y₁) is (x − x₁)² + (y − y₁)² = r².

equals symbol.
An equation is a mathematical expression that contains an equals symbol. Equations often contain algebra. Algebra is used in maths when you do not know the exact number in a calculation.

What is an equation in math example?

An equation is a mathematical sentence that has two equal sides separated by an equal sign. 4 + 6 = 10 is an example of an equation. We can see on the left side of the equal sign, 4 + 6, and on the right hand side of the equal sign, 10.

What is equation in simple way?

A simple equation refers to a mathematical equation that expresses the relationship between two expressions on both sides of the ‘equal to’ sign. This category of an equation consists of a variable, usually in the form of x or y. Solving simple equations often require rearranging it.

What is an equation Class 7?

An equation is a statement of equality between two mathematical expressions containing one or more variables.Now this is called a simple equation. This equation can simplify problems and help us finding the age of Rahul’s father by just substituting Rahul’s age in place of x.

How do you explain an equation to a child?

An equation is a number sentence where one side equals the other, for example: In this case, we know that 4 + 4 = 8 and 10 – 2 = 8, so both sides of this equation are equal, which means it is correct.

Where do you write equations?

Write an equation or formula

• Choose Insert > Equation and choose the equation you want from the gallery.
• After you insert the equation the Equation Tools Design tab opens with symbols and structures that can be added to your equation.

What is equation and expression?

An expression is a number, a variable, or a combination of numbers and variables and operation symbols. An equation is made up of two expressions connected by an equal sign.

What do we use equations for?

An equation is the mathematical representation of those two things which are equal, one on each side of an ‘equals’ sign. Equations are useful to solve our daily life problem. Most of the times we take pre algebra help to resolve real life problems.This is a part of the math equation.

What are the steps to solving an equation?

A General Rule for Solving Equations

1. Simplify each side of the equation by removing parentheses and combining like terms.
2. Use addition or subtraction to isolate the variable term on one side of the equation.
3. Use multiplication or division to solve for the variable.

Is it an equation or expression?

The best way, to identify, whether a given problem is an expression or equation is that if it contains an equal to sign (=) it is an equation. However, if it does not contain an equal to (=) sign, then it is just an expression. It carries numbers, variables and operators, that are used to show the value of something.

What is an equation in maths ks2?

An equation is a sum with an unknown value. Solving an equation means undoing a process to find the unknown value.

What is the symbol for equation?

Basic math symbols

Symbol	Symbol Name	Example
≠	not equal sign	5 ≠ 4 5 is not equal to 4
≈	approximately equal	sin(0.01) ≈ 0.01, x ≈ y means x is approximately equal to y
>	strict inequality	5 > 4 5 is greater than 4
<	strict inequality	4 < 5 4 is less than 5

How do you write a balanced equation?

Count the numbers of atoms of each kind on both sides of the equation to be sure that the chemical equation is balanced.
Adjust the coefficients.

1. 1 Pb atom on both reactant and product sides.
2. 2 Na atoms on reactant side, 1 Na atom on product side.
3. 2 Cl atoms on both reactant and product sides.

Can Word solve equations?

Yes, Word can do this (the Equation Editor is built in; the computation and graphing stuff is part of a free add-in from Microsoft).

How do you write equations in docs?

Insert an equation

1. Open a document in Google Docs.
2. Click where you want to put the equation.
3. Click Insert. Equation.
4. Select the symbols you want to add from one of these menus: Greek letters. Miscellaneous operations. Relations. Math operators. Arrows.
5. Add numbers or substitute variables in the box.

How do you write an equation in one variable?

The linear equations in one variable is an equation which is expressed in the form of ax+b = 0, where a and b are two integers, and x is a variable and has only one solution. For example, 2x+3=8 is a linear equation having a single variable in it.

How do you write an equation given two points?

Steps to find the equation of a line from two points:

1. Find the slope using the slope formula.
2. Use the slope and one of the points to solve for the y-intercept (b).
3. Once you know the value for m and the value for b, you can plug these into the slope-intercept form of a line (y = mx + b) to get the equation for the line.

There are numerous ways in which one may define an equation. In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an ‘equal’ sign. The most basic and simple algebraic equations consist of one or more variables in math. 

Example of an Equation:

4y + 2 = 18

4×2 + 3x − 28 = 0

9m = 49.5

$frac{u}{6}$ = 8

Non examples of an Equation:

k + 7

u + w

x$^{3}$ + 5x

9t

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Different Types of Equations:

Some of the math equations used in algebra are:

1. Linear Equation

A linear equation may have more than one variable. A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation.

1. Quadratic Equation

This is a second-order equation. In quadratic equations, at least one of the variables should be raised to exponent 2.

Example: 

ax$^{2}$ + bx + c = 0

$frac{p^{2}}{9}$ − 1 = 0

1. Cubic Equation

This is a third-order equation. In cubic equations, at least one of the variables should be raised to exponent 3.

Example: 

ax$^{3}$ + bx$^{2}$ + cx + d = 0

a$^{3}$ – 27 = 0

1. Rational Equation

A rational equation is an equation that contains fractions with a variable in the numerator, denominator, or both.

Example:  $frac{x}{2} = frac{x + c}{4}$. 

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Expression vs Equation

A math expression is different from a math equation. An equation will always use an equal (=) operator between two math expressions.

For example,

What is a Solution of an Equation?

The value of the variable which makes the equation a true statement is the solution of the equation.

Example 1:

Verify that x = 3 is the solution of an equation 4x − 8 = − 5 + 3x

Substitute x = 3 in the given equation

LHS

4x − 8 = 4(3) − 8 = 12 − 8 = 4

RHS

−5 + 3x = −5 + 3(3) = −5 + 9 = 4

LHS = RHS

So,  x = 3 is the solution of an equation 4x − 8 = −5 + 3x.

Example 2:

Verify that y = −2 is the solution of an equation 2m – 4 = 1

Substitute y = −2 in the given equation.

LHS

2m – 4 = 2(−2) − 4 = − 4 − 4 = − 8

RHS 

1

− 8 ≠ 1

LHS ≠ RHS

So,  y = −2 is not the solution of given equation 2m – 4 = 1.

How to Solve Linear Equations with One Variable?

• Simplify the expressions inside parentheses, brackets, braces and fractions bars.
• The same quantity can be added, subtracted, multiplied or divided from both sides of an equation without changing the equality. 

Or

Any term of an equation may be taken from one side to the other with the change in its sign. This process is called transposition.

Example:

4a – 9 = 13 – 7a

4a + 7a = 13 + 9 [transpose −7a to LHS and − 9 to RHS]

11a = 22 [add like terms]

a = $frac{22}{11}$ [transpose 11 to RHS]

a = 2

Example:

$frac{1}{5} + 3w = frac{2}{5}$

3w = $frac{2}{5} – frac{1}{5}$ [transpose 15 to RHS]

3w = $frac{1}{5}$

w = $frac{1}{5times 3}$ [transpose 3 to RHS]

w = $frac{1}{15}$

Solved Examples:

Example 1: Solve for x.

x + 8 = 12

Solution:

Here is the equation to solve: x + 8 = 12

We need to leave x alone on one side of the equation. For this, we must take 8 out of both sides.

So, x + 8 – 8 = 12 – 8

or, x = 4

Example 2: Determine if the value 3 is a solution of the equation:

4x – 2 = 3x + 1

Solution:

We will substitute the value of 3 in this equation and will check if the left-side equation is equal to the right-hand side.

So, 

4(3) – 2 = 3(3) + 1

or, 12 – 2 = 9 + 1

or, 10 = 10

Yes, 3 is a solution to the given equation.

Example 3: Solve the equation: 6(2x + 3) + x – 7 = 3(5x + 7) + 2x

Solution:

6(2x + 3) + x – 7 = 3(5x + 7) + 2x

Expanding the terms we get,

12x + 18 + x – 7 = 15x + 21 + 2x

or, 13x + 11 = 17x + 21

On further simplification,

13x – 17x = 21 – 11

−4x = 10

x = -$frac{10}{4}$

x = -$frac{5}{2}$

Practice Problems

7x + 5y = 19

5 — 2

$frac{4}{7} — frac{2}{7}$

3a + 9b

Correct answer is: 7x + 5y = 19
As option a has an equal sign (=) between two math expressions its an equation. The other options are expressions.

n + 2 = 9

7 — g = 0

x — 4 = 3

h$times$ 1 = 8

Correct answer is: h$times$ 1 = 8
7$times$ 1 = 7 and 7 ≠ 8. So 7 is not the solution for the given equation.

3

2

−3

−1

Correct answer is: −3
9k = −27
k = -$frac{27}{9}$
k = -3

Conclusion

We have thus learned the definition of the equation and its different types. Moreover, a few questions have also been solved here to give a clear idea to the students about solving an equation. A student can have a strong grip on this concept by practising such problems.Teaching maths concepts can be challenging, especially when the students are young kids. So, to make parents’ and teachers’ lives easy, SplashLearn has brought some courses specifically designed for students from K-8. After all, it should be fun to learn!    

Frequently Asked Questions

What are the types of Equation?

There are three types of equations based on the degree. Linear equation, quadratic equation, and cubic equation.

How are linear equations used in daily life?

Linear equations are used to find the wages based on hourly pay rate, speed, and medicine dosage based on the patient’s weight.

What must an equation have?

An equation in algebra is a statement of equality that contains one or more unknown quantities or variables.

How do you solve equations?

Solving Equations – A General Rule

• Remove parentheses and combine like terms on each side of the equation.
• To isolate the variable term, you can use addition or subtraction.
• To solve for a variable, use multiplication or division.

Algebraically equality of two objects is often defined if they are «the same». We define this in different contexts, but mostly we define it together with the concept of an order relation. Some examples:

In set theory:
$$ A subseteq B land B subseteq A implies A=B$$

Since a set is defined by the elements it contains, if one set is contained in the other and vice versa, clearly they must contain the same elements. Here we see equality because a set is uniquely determined by its elements.

In ordered fields, using the trichotomy axiom ($mathbb{Q}$ is an ordered field, not completely ordered though, $mathbb{R}$ is):
$$ aleq b land bleq a implies a=b$$

In group theory: let $e$ be the identity element and $a$ an arbitray element in the group:
$$ e cdot a = a cdot e=a$$
This statements tells us that applying the identity elements does not do anything, it also does not matter from which side we apply it. All these examples only further support the idea we see equality as two things being «the same». We can even define the relation «=» for any two numbers, clearly:

$$ a=a$$
$$ a=b implies b=a$$
$$ a=b land b=c implies a=c $$
So equality is an equivalence relation.

Another fun example there are equally as much even-sized subsets as odd-sized of a given set:

$$ 0=(1-1)^n = sum_{k=0} ^n binom{n}{k} (-1)^n $$
Here the binomial tells us how many $k-sized$ subsets there are of a set $n$, we notice that all possible subsets are either even or odd by a partition of the natural numbers into these two sets. We can show there exists a bijection between the two so they have the same cardinality. We can split the sum up over odd and over even $k$ and notice that these two terms cancel, there must be the same amount of terms and thus there must be the same amount of subsets for all $n$. So there are equally as much off-subsets as even -subsets.

Surprisingly when two objects are «the same» in algebra, but have a different representation, we speak of the idea of an «isomorphism», which is also an interesting concept. This means we can identify both the structure and the elements uniquely. All very interesting ideas, definitely a nice question to think about, thank you!

A mathematical formula is a type of equation as all formulae have an equal sign. However, they are equations with a specific purpose. They give us a way of working something out. For example, if we wanted to convert degrees Fahrenheit to degrees Celsius, we could use a formula. There are many formulae that are specifically useful for GCSE mathematics, including the quadratic formula, the trigonometric formulae, and also the speed, distance, and time formula. In the example below we will look at some specific formulae.

Asked by: Emely Stanton

Score: 4.8/5
(41 votes)

An equation is simply a statement in math in which two things on equal. There are two expressions, one on each side of an equals sign. For example: X = 7. In this case, another way to write the equation would be 7 = 7.

What is an example of an equation in math?

An equation is a mathematical sentence that has two equal sides separated by an equal sign. 4 + 6 = 10 is an example of an equation. We can see on the left side of the equal sign, 4 + 6, and on the right hand side of the equal sign, 10.

What is equation and examples?

In algebra, an equation can be defined as a mathematical statement consisting of an equal symbol between two algebraic expressions that have the same value. … For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an ‘equal’ sign.

What is a number equation?

An equation is simply a number sentence with a variable in the place of an unknown value. A variable is a letter that is substituted in place of the unknown value. It can be any letter you want, although x is most commonly used.

What is a simple equation?

What is a Simple Equation? In the simplest of terms, a variable and an ‘equal to’ sign make up a simple equation. Therefore, a simple equation is a mathematical representation of two expressions on either side of an ‘equal to’ sign. It mostly consists of a variable, frequently accompanied by a numerical constant.

45 related questions found

In mathematics, you must have come across the word ‘equations’. You must have read questions like, ‘Solve the equation 2x + 12 = 33 to find the value of x’. But, what is the meaning of the term equation? Equations, in math, are statements. These statements often have two linked expressions with an ‘equal to sign. Equations can even have more than two expressions. Mathematicians have solved some equations having more than two expressions up to a certain level. Beyond that, even they are rendered helpless. But, we don’t have to worry about those difficult equations. In this article, we will learn the fundamental concepts of every equation known to humanity. We will study what an equation is and how many types of equations are there in this world.

What are Equations? 

A mathematical statement made up of an equal sign between two algebraic expressions with the same value is known as an equation. One or more variables are present in even the basic and most common algebraic equations in mathematics. 

Equations Definition

Equations are very easy to identify. Wherever you see two algebraic expressions on the left and right-hand sides of the equal ‘=’ sign, you have found an equation. So we can say that statements with the ‘equal to’ sign in between them are known as equations. If the equal to symbol is not present between the expressions, they are not considered an equation. In the next section of this article, we will see the difference between equations and expressions. For now, look at the examples given below:

Example 1: Is x + y = 13 an equation?

Solution: Yes, x + y = 13 is an equation because it has an ‘equal to’ sign between x + y and 13.

Example 2: Is a – 24 + 13y an equation?

Solution: No! a – 24 + 13y is not an equation because it doesn’t have an equal sign.

Example 3: Is 66 – 12 = 34 – 3 an equation?

Solution: Indeed, this is an equation because 66 – 12 and 34 – 3 have the ‘equal to’ sign between them.

Equations are used in mathematics to solve the values of unknown quantities. They are used in solving word problems and many daily life problems involving time and distance, work, profit and loss, and many more. Let us now learn the basic difference between an equation and an expression.

Equation Vs. Expression

Equations and Expressions are two different things. Students often confuse the two and make a lot of mistakes. See the table mentioned below to clarify the difference between an equation and an expression.

Equation	Expression
In math, an equation is formed when two expressions have the same value and are put together with an ‘equal to’ symbol in the middle.	It’s a mathematical statement with at least one word and several terms linked by operators.
It has a “=” symbol next to it.	An equal to “=” symbol does not appear in an expression.
It can be solved to determine the unknown quantity’s value.	Maximum it can be reduced to its most basic form.
For example: 7x – 2a = 34, 4a + 22 = 3c + b, etc.	For example: 34x – 3z + 2y, a + 3k, etc.

Parts of an Equation

In every equation, the left-hand side should be equal to the right-hand side. Both sides must be equal. Coefficients, variables, operators, constants, terms, expressions, and an ‘=’ sign are all essential components of an equation. An equation may have any one or all of these terms revolving around the ‘=’ sign. Let us learn these terms individually:

• Term: Any numeric or algebraic entity present around the operators is known as a term. 
• Variables: All the alphabetic terms present in an equation whose value is unknown are known as variables.
• Constants: All the numeric terms of an equation are known as constants.
• Coefficients: The constant terms present just before the variables are coefficients. 
• Operators: The arithmetic operators like the addition sign, subtraction sign, equal sign, etc., are known as operators.

An equation may have only numeric terms, algebraic terms, or both. We shall learn the steps to solve an equation in the next module.

How to Solve Equation?

You can imagine an equation as a weighing scale. Both sides of an equation result in the same value, hence keeping the weighing scale in balance. It still holds whether we add or remove the same amount from both sides of an equation. The same remains true whether we multiply or divide the same integer into both sides of an equation.

Whatever operation you perform on the left side must also be implemented on the right side of an equation. This is done to maintain the balance of the equation. Let us take an example to understand this concept.

Consider an equation x + 3y = 2a + b. This is a balanced equation. Let us say that we want to add 20 to the left-hand side. The left-hand changes to x + 3y + 20. But now, the equation has been disturbed. We need to add 20 to the right side to balance it. Therefore the new equation will be x + 3y + 20 = 2a + b + 20.

Let us now learn how to solve an equation:

• Step 1: Always keep the variables to one side of the ‘=’ symbol and constants to the other side of the ‘=’ symbol. Normally, the variables are kept to the left-hand side and constants on the right-hand side of the ‘=’ sign. 

• Step 2: Do the necessary operations on the like terms like addition or subtraction. Do not operate like terms with unlike terms. You will be following the wrong approach if you do so.

• Step 3: Simplify the equation and get the desired answer. 

Look at the example given below:

Example 1: Solve the equation a + 34b – 22 = b + 18

Solution: 

From step 1: a + 34b – b = 18 + 22

From step 2: a + 33b = 40

From step 3: This equation cannot be further simplified hence a + 33b = 40, is the solution of this equation. 

Example 2: Solve the equation 4x + 3 = x + 27

Solution: 

From step 1: 4x – x = 27 – 3

From step 2: 3x = 24

From step 3:  x = 24/3, therefore the value of x and the solution to this equation is 8.

Types of Equation

The equations are classified based on the degree of variables. A degree means the power assigned to any variable. Variables having a degree of 1 are known as linear variables. Similarly, the variables having a degree of two are known as quadratic variables, and a degree of three are known as cubic variables, and so on. The equations are classified as linear equations, quadratic equations, cubic equations, etc. Let us study them in detail.

Linear equations: The equations having all the variables of degree as one are known as linear equations. For example, x + 2y = 11, x + 5 = 0, etc. These are further classified as linear equations in one variable, linear equations in two variables, and so on. The standard form of a linear equation is kx + ly – m = 0, where x and y are variables and k, l, and m are constants. 

Quadratic equations: The equations having at least one variable of degree two are known as quadratic equations. For instance, x² + a = 22, k² – m² – n = 0, etc. kx² + lx + m = 0 is the standard form of a quadratic equation. 

Cubic equations: The equations having at least one variable having a degree of 3 are known as cubic equations. For example, x³ – x² + 2 = 31y , k³ – m² + n³ = 19, etc. The standard form to write a cubic equation is kx³ + lx² + mx + n = 0. 

In all the above cases k, the coefficients of x, x², and x³ can never be zero. If k = 0, then the terms containing the degree will be eradicated, making the equation invalid.

For instance if in the equation kx² + lx + m = 0, if k = 0, then the equation becomes lx +m = 0, which is not a quadratic but a linear equation. 

Key Notes on Equations

The solution or root of the equation refers to the values of the variable that make an equation true.

When the same number is summed, subtracted, multiplied, or divided into both sides of an equation, the solution remains unaltered.

A straight line graphs a linear equation in one or two variables, while a parabola graphs a quadratic equation.

Practice Questions 

Which is a solution of the equation x + 4 = 6? 

(1) x = 2 

(2) x = 3 

(3) x = 4 

(4) x = 6 

2. The answer to the equation 3n – 2 = 46 is n: 

(1) 12 

(2) 11 

(3) 16 

(4) none of these 

3. The solution to the equation 3p + 4 = 25 is p =  

(1) 5 

(2) 6 

(3) 4 

(4) 7 

Frequently Asked Question?

1. How do you write a complete Equation?

You can write a complete equation by adding the equal sign to the beginning of an equation and then writing all of the variables, numbers, or symbols that appear on either side of it.

For example, x + 2 = 5 In this example, “x” is on one side of the equal sign and “2” is on the other side.

2. How do you write a complete Equation?

In order to write a complete equation, you need to know how to solve for one variable at a time.

Let’s say we have the equation:

x + y = 12

To solve for x, we need to isolate it by subtracting y from both sides of the equation:

x + y – y = 12 – y

x = 12 – y

Now, if we want to solve for y, then we just do the same thing: subtract x from both sides and then divide by 2.

3. How does one solve the Equation?

Let’s start with a simple equation:

x² + 3x + 2 = 0

To solve this equation, we’re going to need to isolate the x term. We can do that by adding or subtracting multiples of the same number to both sides of the equation. We will make our life easier by adding 6 to both sides of the equation. This gives us:

x² + 3x + 2 = 0 + 6 (add 6)

x² + 3x + 8 = 6 (subtract 6)

We now have an equation where we can isolate our x term:

(x² + 3x) – 8 = 0 (subtract 8 from both sides)

We can now solve for x: divide both sides by (x² + 3x):

1/2(x² + 3x) – 8/2 = 0 => 1/2(1+3*(-8)) -8/2 => 1/2(-11) -8/2 => 1/2(-11) – (-8/2) => 1/2(-11 – 16) => 1/2(-17)=> x=-17

4. What are the steps in an Equation?

 An equation is a mathematical statement that two things are equal. There are four steps to solve an equation:

1. Find the value of one side of the equation by itself. For example, if you have 3+2=5, then 5=3+2.

2. Solve the variable in terms of other variables in the equation by subtracting or dividing both sides of the equation by something. For example, if you have 3x+2=5, then 3x-5/3=0 or -5/3 x = 0, so x = -1/3 or 1/3.

3. Substitute your answer back into the original equation and solve for another variable in terms of all others; this will give you another solution to check against. For example, if x = 1/3 and y = -1/3 (from step 2), then 3(-1/3)+2(-1/3)=5(-1/3). If you add these together, you get -4/9 = 5(-1/9) and -4 + 9 = 5, so -4 + 9 = 5; therefore x may be 1/3 but not necessarily 1/9! So we’re done here!

5. What are the types of Equations?

There are several different types of equations:

* Linear equations are equations that can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

* Quadratic equations have the form ax^2 + bx + c = 0.

* Absolute value equations are of the form |x|.

* Rational expressions are of the form p/q, where p and q are polynomials with no common factors and q ≠ 0.