Elimination by subtraction

Objective. Eliminate one unknown by subtracting equations.

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Learn: the key idea

Elimination by subtraction solves simultaneous equations by subtracting one equation from the other to cancel a variable with matching coefficients and the same sign. First confirm one variable has identical coefficients in both equations; if needed, multiply through to create a matching pair before subtracting. Subtracting removes that variable, leaving a single equation to solve, after which the found value is substituted back into either original equation. For example, with 5x + 2y = 16 and 3x + 2y = 10, subtracting gives 2x = 6, so x = 3, and substituting gives y = 0.5.

Worked example

Solve the simultaneous equations 5x + 3y = 21 and 5x + y = 13 by elimination.

1 $Both equations have matching 5x terms, so subtract to eliminate x.$
2 $Subtracting: (5x - 5x) + (3y - y) = 21 - 13, giving 2y = 8.$
3 $Divide by 2: y = 4.$
4 $Substitute y = 4 into 5x + y = 13: 5x + 4 = 13, so 5x = 9 and x = 1.8.$

Answer: x = 1.8, y = 4

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Common mistake. Pupils often subtract the equations in an inconsistent order, computing (first − second) for the left side but (second − first) for the right side, which produces a sign error. Subtract both sides of the equation in the same consistent direction.

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