# Linear Equations Definition, Formula, Examples | What is a Linear Equation?

Linear equations are the first-order equations. The highest power of variables is 1 in the linear equation. These equations are defined as the straight lines in the coordinate geometry. The representation of straight-line equation is y = mx + c. Learn about the linear equations in one variable, two variables, and three variables and solving linear equations from the following sections.

## Forms of Linear Equation

Some common forms of the linear equations are mentioned below:

General Form of Linear Equation:

Generally, linear equations are a combination of variables and constants. The standard form in one variable is ax + b = 0. Where a ≠ 0 and x is the variable. The standard form of a linear equation in two variables is ax + by + c = 0, Where a ≠ 0, b ≠ 0, x, and y are variables. The standard form of a linear equation in three variables is ax + by + cz + d = 0. Where a ≠ 0, b ≠ 0, c ≠ 0, x, y, z are variables.

Slope Intercept Form:

It is the most common form of the linear equation, which is represented as y = mx + c.
where x, y are the variables, m is the slope of the line, c is the intercept.

Point Slope Form:

Here the straight-line equation is formed by considering the points in the xy plane. such that (y – y₁) = m(x – x₁)

Where (x₁, y₁) are the coordinates of the line.

We can also express the point-slope form as y = mx + y₁ – mx₁.

Intercept Form:

A line that is neither parallel to axes of the coordinate plane nor passes through the origin but intersects the axes in two different points represents the intercept form. The intercept values x₀, y₀ are non zero and form the equation as x / x₀ + y / y₀ = 1

Two-Point Form:

If a line passes through the two points (x₁, y₁) and (x₂, y₂), then that equation of the line is given as y – y₁ = [(y₂ – y₁) / (x₂ – x₁)](x – x₁)

Here (y₂ – y₁) / (x₂ – x₁) is the slope of the line and x₁ ≠ x₂.

### How to Solve Linear Equations?

We have given the step by step detailed explanation of how to solve the linear equation in one variable. Have a look at the following points and solve them easily. The first rule is both sides of the equation should be balanced. The equality sign between the expressions represents that both are equal. To balance that equation, we need to perform certain mathematical operations on both sides that should not change the actual equation.

Solution of Linear Equations in Two Variables

You can follow any of these methods to solve the linear equations in two variables. They are the method of substitution, method of elimination, cross multiplication method, and determinant methods

Solution of Linear Equations in Three Variables

To solve the linear equations with 3 variables, you need to follow the matrix method.

### What is transposition?

Any term of an equation can be shifted to the other side with a change in its sign without affecting the equality. This process is called transposition.

While transposing a term from one side to the other side of equality you need to follow the below instructions

• We can simply change its sign and carry it to the other side.
• ‘+’ sign changes to ‘-‘ sign and vice-versa.
• ‘x’ sign of a term changes to ‘÷’ sign and vice-versa.
• Simplify the left-hand side expression such that each side contains one term.
• Finally, simplify the equation to get the variable value.

### Linear Equations Questions with Solutions

Example 1.

Solve the linear equations in one variable.

(a) x + 8 = 56

(b) x – 5 = 20

Solution:

(a) x + 8 = 56

Subtract 8 from both sides.

x + 8 – 8 = 56 – 8

x = 48

(b) x – 5 = 20

x – 5 + 5 = 20 + 5

x = 25

Example 2.

(a) x + ⅓ = 25

(b) 3x – 6 = 9

Solution:

(a) x + ⅓ = 25

Subtract ⅓ from both sides.

x + ⅓ – ⅓ = 25 – ⅓

x = (25 x 3 – 1) / 3

x = (75 – 1) / 3

x = 74 / 3.

(b) 3x – 6 = 9

3x – 6 + 6 = 9 + 6

3x = 15

Divide 3 by both sides.

3x / 3 = 15 / 3

x = 5.

Example 3.

Solve the following linear equations.

(a) 0.5x = 0.3

(b) 2x = 12

Solution:

(a) 0.5x = 0.3

Divide both sides by 0.5

0.5x / 0.5 = 0.3 / 0.5

x = 0.6

(b) 2x = 12

Divide both sides by 2

2x / 2 = 12 / 2

x = 6.