# Linear Equations – Definition and How to Solve

Solving equations is probably the most important and useful topic in algebra. We can solve a linear equation by applying operations to both sides of the equation so that we completely isolate the variable.

In this article, we will learn about linear equations. We will also learn how to solve these equations and look at solved examples to improve the retention of the concepts.

##### ALGEBRA

Relevant for…

Learning how to solve linear equations.

See examples

##### ALGEBRA

Relevant for…

Learning how to solve linear equations.

See examples

## Definition of linear equations

Linear equations are defined as the equations in which the maximum power of the variables is 1. These equations are called linear because they produce a straight line when graphed.

The following are examples of linear equations and nonlinear equations:

### EXAMPLES

Linear equation:

• $latex 4x+2=10$
• $latex \frac{1}{3}x=9$
• $latex 3x+2=-2x+1$
• $latex 4(x-2)+2=-2x+1$

Nonlinear equation:

• $latex {{x}^{2}}+x-3=2$
• $latex 2{{x}^{2}}-3{{x}^{2}}=6$
• $latex \frac{1}{2}{{x}^{2}}=2x+1$

## How to solve linear equations?

To solve linear equations, we have to remember that we can perform any operation on one side of the equation, as long as we perform the same operation on the opposite side of the equation.

The following steps are a good method we can use to solve equations:

1. Simplify each side of the equation by removing parentheses and combining like terms.

2. Use addition and subtraction to solve for terms with variables on one side of the equation.

3. Use multiplication and division to solve the equation.

## Linear equations – Examples with answers

### EXAMPLE 1

• Solve the equation $latex 3x+5=14$ for $latex x$.

Solution:

1. Simplify: it is already simplified.

2. Solve for the variables: we move the 5 to the right:

$latex 3x+5-5=14-5$

$latex 3x=9$

3. Solve the equation: we divide by 3:

$latex \frac{3}{3}x=\frac{9}{3}$

$latex x=3$

### EXAMPLE 2

• Solve the equation $latex 3(x-5)-x=-7$ for $latex x$.

Solution:

1. Simplify:

$latex 3(x-5)-x=-7$

$latex 3x-15-x=-7$

$latex 2x-15=-7$

2. Solve for the variables: we move the -15 to the right:

$latex 2x-15+15=-7+15$

$latex 2x=8$

3. Solve the equation: we divide by 2:

$latex \frac{2}{2}x=\frac{8}{2}$

$latex x=4$

### EXAMPLE 3

• Solve $latex 3(x-1)+2x=x+5$ for $latex x$.

Solution:

1. Simplify:

$latex 3(x-1)+2x=x+5$

$latex 3x-3+2x=x+5$

$latex 5x-3=x+5$

2. Solve for the variables: we move the -3 to the right and the x to the left:

$latex 5x-3+3=x+3+5$

$latex 5x=x+8$

$latex 5x-x=x+8-x$

$latex 4x=8$

3. Solve the equation: we divide by 4:

$latex \frac{4}{4}x=\frac{8}{4}$

$latex x=2$

### EXAMPLE 4

• Solve the equation $latex 4(x-2)+5=2(x+5)-7$ for $latex x$.

Solution:

1. Simplify:

$latex 4(x-2)+5=2(x+5)-7$

$latex 4x-8+5=2x+10-7$

$latex 4x-3=2x+3$

2. Solve for the variables: we move the -3 to the right and the 2x to the left:

$latex 4x-3+3=2x+3+3$

$latex 4x=2x+6$

$latex 4x-2x=2x+6-2x$

$latex 2x=6$

3. Solve the equation: we divide by 2:

$latex \frac{2}{2}x=\frac{6}{2}$

$latex x=3$

### EXAMPLE 5

• Solve the equation $$x+14=4(x-5)+2(x+1)+7$$ for $latex x$.

Solution:

1. Simplify:

$$x+14=4(x-5)+2(x+1)+7$$

$latex x+14=4x-20+2x+2+7$

$latex x+14=6x-11$

2. Solve for the variables: we move the 14 to the right and the 6x to the left:

$latex x+14-14=6x-11-14$

$latex x=6x-25$

$latex x-6x=6x-25-6x$

$latex -5x=-25$

3. Solve the equation: we divide by -5:

$latex \frac{-5}{-5}x=\frac{-25}{-5}$

$latex x=5$