Simultaneous Equations

What they are

Two equations, two unknowns. The solution is the pair (x, y) that satisfies both — graphically, where the two lines cross.

Elimination method (most reliable)

Solve 2x + y = 7 and 3x − y = 8

2x + y = 7
3x − y = 8        (y has opposite signs → ADD to eliminate)
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5x     = 15   →   x = 3

Substitute back: 2(3) + y = 7 → y = 1. Solution: x = 3, y = 1.

When coefficients don't match

Solve 3x + 2y = 12 and x + y = 5.

Multiply the second by 2: 2x + 2y = 10. Now subtract:

3x + 2y = 12
2x + 2y = 10     (same sign → SUBTRACT)
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 x      = 2   →   y = 3

Rule for add vs subtract

• Matching coefficients with the same sign → subtract.
• Matching coefficients with opposite signs → add.

("Same Sign Subtract.")

Substitution method

Best when one equation is y = …. Rearrange, then substitute into the other equation.

Exam tip

Check in BOTH original equations — a fast way to catch arithmetic slips. Always state both x and y.