Simultaneous equations

Simultaneous equations are two equations with two unknowns that are both true at the same time. Eliminate one letter by adding or subtracting the equations, solve for the other, then substitute back.

Key facts

  • Make the number in front of one letter the same in both equations.
  • Same signs: subtract. Different signs: add.
  • Substitute your answer back into one equation to find the other letter.
  • Check both equations at the end.

Worked examples

Example 1

Solve x + y = 10 and x − y = 4.

  1. Add the equations: the y’s cancel. 2x = 14
  2. Solve. x = 7
  3. Substitute into the first. 7 + y = 10, y = 3

Answer: x = 7, y = 3

Example 2

2 coffees and 3 teas cost £11. 2 coffees and 1 tea cost £7. Find the price of a tea.

  1. Write equations. 2c + 3t = 11, 2c + t = 7
  2. Subtract: the coffees cancel. 2t = 4
  3. So t = 2

Answer: £2