Solve simultaneous equations

A standard method from Number and algebra basics: the steps and worked examples.

Steps

Choose the method:
- substitution, if one equation already gives x or y alone;
- elimination, otherwise. Multiply so one unknown has the same coefficient, then add or subtract.
Solve for one unknown.
Substitute back to find the other.
Check in both equations.

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Worked examples

Example 1: Solve the simultaneous equations 2x + 3y = 12 and x − y = 1 (substitution).

x = y + 1                      from the second equation
2(y + 1) + 3y = 12             substitute
       5y + 2 = 12
            y = 2,  x = 3      check: 2(3) + 3(2) = 12 ✓, 3 − 2 = 1 ✓

Example 2: Solve the simultaneous equations 3x + 2y = 16 and 5x − 2y = 8 (elimination).

(3x + 2y) + (5x − 2y) = 16 + 8      add: the y-terms cancel
                   8x = 24
                    x = 3
3(3) + 2y = 16                      substitute back
       2y = 7
        y = 3.5                     check: 5(3) − 2(3.5) = 15 − 7 = 8 ✓

For the whole topic, read Number and algebra basics: key points, common mistakes and how the DSE tests it.

Steps

Worked examples

Other methods in this topic