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# Linear Equations: Formula & Examples

## What is a linear equation?

A linear equation is any equation that takes the form: ax + b = 0, where a and b are constant numbers and x is a variable. You may be more familiar with it written as y = mx + b, where m and b are constants and x and y are variables.
Linear equations are a little easier to solve than quadratic equations, and you solve them in a different way. Since a quadratic equation is any equation that can be written in the form of ax2 + bx + c = 0, if the value of a is zero, then the equation becomes a linear equation, not a quadratic equation.

The general form of a linear equation:

– One variable is ax + b = 0, where a ≠ 0 and x is the variable.

– Two variables is ax + by + c = 0, where, a ≠ 0, b ≠ 0 , x and y are the variables.

– Three variables is ax + by + cz + d = 0 where a ≠ 0, b ≠ 0, c ≠ 0, x, y, z are the variables.

### How to solve Linear Equations?

For the equations involving 2 variables, we need to have two equations to solve for the two variables. There are various methods to solve these equations. Some of them are as follows:

• Cross multiplication method
• Method of substitution
• Method of elimination
• Matrix method
• Determinant methods

## Linear equations in one variable

Linear Equations are those algebraic expression is equated to a particular constant or an algebraic expression. In a linear equation there will be definite values for the variables to satisfy the condition of the equation.

For example, Solution for the system of linear equations

x + y = 12,

2x + 3y = 32

x = (4, 8)

## Linear equations in two variables

Linear equations of two variables are of the form, ax + by + c = 0. To solve for the variables x and y, we need to have two equations. Otherwise for every values of x there will be a corresponding value of y.  Hence a single equation has infinite number of solutions and each solution is the point on the line.

### Solved Example

Question: Solve x + y = 5
Solution:
We have x + y = 5

=> y = 5 – x

when x = -3, y = 5 – (-3) = 5 + 3 = 8 , the solution is ( -3, 8)

when x = 0,  y = 5 – 0 = 5 , the solution is ( 0, 5)

When x = 5, y = 5 + 5 = 10, the solution is ( 5, 0)

## Graphing Linear Equations

• Graph of linear equations of one variable is a point on the real number line.

For Example: Draw graph for the linear equation, 2x + 5 = 9

Given 2x + 5 = 9

=> 2x + 5 – 5 = 9 – 5

=> 2x = 4

=> 2×2=42

=> x = 2

This can be represented on the real number line as follows: In the above number line we can see that the solution, x = 2 is marked.

• Graph of linear Equations of 2 variables will be a straight line, which can be shown on co-ordinate graph.

Let us draw the graph for the equation, x + y = 5

We have the points, (-3, 8), (0, 5), (5, 0) 