Solving linear equations

In order to solve any equation, we need to get the variable we are solving for on one side of the equation and everything else on the other side.

To do this, we follow one simple rule:

Whatever you do to one side of the equation, you must do to the other side.

With that rule in mind, we can just use logic to think ‘what do I need to do in order to get the variable on its own?’

Note: for a note explaining how to rearrange equations, see rearranging-equations.

Example questions

Solve for x: 2x + 3 = 7

• Subtract 3 from both sides:
  • 2x + 3 - 3 = 7 - 3
  • 2x = 4
• Divide both sides by 2:
  • \frac{2x}{2} = \frac{4}{2}
  • x = 2
• Answer: x = 2

Solve for y: 5y - 10 = 15

• Add 10 to both sides:
  • 5y - 10 + 10 = 15 + 10
  • 5y = 25
• Divide both sides by 5:
  • \frac{5y}{5} = \frac{25}{5}
  • y = 5
• Answer: y = 5

Solve for z: 3z + 4 = 2z + 9

• Subtract 2z from both sides:
  • 3z - 2z + 4 = 2z - 2z + 9
  • z + 4 = 9
• Subtract 4 from both sides:
  • z + 4 - 4 = 9 - 4
  • z = 5
• Answer: z = 5

Solve for a: \frac{4a}2 + 6 = 10

• Subtract 6 from both sides:
  • \frac{4a}2 + 6 - 6 = 10 - 6
  • \frac{4a}2 = 4
• Multiply both sides by 2:
  • 2 \times \frac{4a}2 = 4 \times 2
  • 4a = 8
• Divide both sides by 4:
  • \frac{4a}{4} = \frac{8}{4}
  • a = 2
• Answer: a = 2

Solve for b: 7 - 3b = 1 + 2b

• Subtract 1 from both sides:
  • 7 - 1 - 3b = 1 - 1 + 2b
  • 6 - 3b = 2b
• Add 3b to both sides:
  • 6 - 3b + 3b = 2b + 3b
  • 6 = 5b
• Divide both sides by 5:
  • \frac{6}{5} = \frac{5b}{5}
  • \frac{6}{5} = b
• Answer: b = \frac{6}{5}

Solve for c: 2(c - 3) + 4 = 10

• Expand the left side:
  • 2c - 6 + 4 = 10
  • 2c - 2 = 10
• Add 2 to both sides:
  • 2c - 2 + 2 = 10 + 2
  • 2c = 12
• Divide both sides by 2:
  • \frac{2c}{2} = \frac{12}{2}
  • c = 6
• Answer: c = 6

Solve for d: \frac{d + 2}3 = 5

• Multiply both sides by 3:
  • 3 \times \frac{d + 2}3 = 5 \times 3
  • d + 2 = 15
• Subtract 2 from both sides:
  • d + 2 - 2 = 15 - 2
  • d = 13
• Answer: d = 13

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