30 min read

Simultaneous Equations

Two equations, two unknowns: elimination and substitution both find the one pair of values that fits both.

What Are Simultaneous Equations?

One equation with two unknowns (like x + y = 10) has many possible answers. Two equations together usually pin down exactly one pair of values. Two methods find that pair: elimination and substitution .

Elimination adds or subtracts the two equations so that one letter cancels out completely.

Example 1 — easy numbers, elimination

Solve: x + y = 10 and x - y = 2 . 1 Add the two equations together. The y -terms cancel: (x + y) + (x - y) = 10 + 2 2 2 x = 12, so x = 6 3 Substitute x = 6 into the first equation: 6 + y = 10, so y = 4 Check: 6 - 4 = 2 ✓ Answer: x = 6, y = 4

Example 2 — matching coefficients first

Solve: 2 x + y = 11 and x + y = 7 . 1 The y -coefficients already match (both are 1), so subtract the equations: (2 x + y) - (x + y) = 11 - 7 2 x = 4 3 Substitute x = 4 into x + y = 7 : 4 + y = 7, so y = 3 Check: 2 (4) + 3 = 11 ✓ Answer: x = 4, y = 3

Example 3 — scaling one equation first

Solve: 3 x + y = 14 and x + y = 6 . 1 The y -coefficients already match, so subtract: (3 x + y) - (x + y) = 14 - 6 2 2 x = 8, so x = 4 3 Substitute into x + y = 6 : 4 + y = 6, so y = 2 Check: 3 (4) + 2 = 14 ✓ Answer: x = 4, y = 2

Substitution rearranges one equation to make a letter the subject, then substitutes it into the other equation.

Example 1 — substitution

Solve: y = x + 3 and 2 x + y = 15 . 1 The first equation already gives y in terms of x . 2 Substitute into the second equation: 2 x + (x + 3) = 15 3 Simplify: 3 x + 3 = 15, so 3 x = 12, so x = 4 4 Substitute x = 4 into y = x + 3 : y = 7 Check: 2 (4) + 7 = 15 ✓ Answer: x = 4, y = 7

Practice

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For x + y = 9 and x - y = 3, what should you do to eliminate y ?

Solve x + y = 9 and x - y = 3 . What is x ?

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