# Solve Linear Equations

A Linear Equation is an algebraic equation composed of a constant (e.g. x = 3) or a variable to the power of 1 (e.g. 2a + 3 = 13).

Remember that 2a means 2 × a.

Solving an equation is to figure out what number is represented by the variable. For example, in the equation 2a + 3 = 13, the variable a is 5.

Some hints to solve are:

• Do the opposite operation to what is shown in the equation.
• Move variables (letters) to the left side of the equation.
• Move constants (numbers on their own) to the right side of the equation.
• If you change the side, you change the sign.

## Example One

Solve this linear equation with the variable on one side.

x + 3 = 10
x = 10 – 3
x = 7

## Example Two

Solve this linear equation with the variable on one side.

4x + 3 = 11
4x = 11 – 3
4x = 8

 x = 8 4

x = 2

## Example Three

Solve this linear equation with the variable on one side.

3a – 5 = 7
3a = 7 + 5
3a = 12

 a = 12 3

a = 4

Q1. 5x + 6 = 16
Q2. 9x – 2 = 25
Q3. –3x = –21
Q4. 3x – 5 = –1
Q5. –2x + 8 = 16

A1. 2
A2. 3
A3. 7
A4. –2
A5. –4

## Example Four

Solve this linear equation with the variable in brackets on one side.

2 ( x + 3 ) = 16
2x + 6 = 16
2x = 16 – 6
2x = 10

 x = 10 2

x = 5

## Example Five

Solve this linear equation with the variable in brackets on one side.

4 ( x – 2 ) = 20
4x – 8 = 20
4x = 20 + 8
4x = 28

 x = 28 4

x = 7

## Questions - Solve

Q1. 5 ( x + 2 ) = 25
Q2. 6 ( x + 4 ) = 42
Q3. 9 ( x + 5 ) = 54
Q4. 5 ( x – 2 ) = 5
Q5. –2 ( x + 8 ) = –4

A1. 3
A2. 3
A3. 1
A4. 3
A5. 10

## Example Six

Solve this linear equation with the variable on both sides.

Remember: Move variables (letters) to the left side of the equation. Move constants (numbers on their own) to the right side of the equation. If you change the side, you change the sign.

4x + 8 = 2x + 14
4x – 2x = 14 – 8
2x = 6
x = 3

## Example Seven

Solve this linear equation with the variable on both sides.

5x – 3 = 3x + 11
5x – 3x = 11 + 3
2x = 14
x = 7

## Questions - Solve

Q1. 7x – 6 = 3x + 10
Q2. 9x + 5 = 3x + 23
Q3. 10x + 4 = 5x – 6
Q4. x + 2 = 2x + 5
Q5. 5x = 2x – 9

A1. 4
A2. 3
A3. –2
A4. –3
A5. –3

## Example Eight

Solve this linear equation with fractions.

Multiply both fractions by a number that will cancel out the fractions' denominator. In this example, we will multiply both fractions by 10.

 x = ( x + 3 ) 2 5

5x = 2 ( x + 3 )
5x = 2x + 6
5x - 2x = 6
3x = 6
x = 2

## Question - Solve

 x = ( x – 4 ) 3 6   