Linear simultaneous equations

Objective. Use elimination and substitution with non-unit coefficients.

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Learn: the key idea

Linear simultaneous equations with non-unit coefficients are solved by elimination or substitution to find values of two variables satisfying both equations at once. For elimination, multiply one or both equations so a variable has matching coefficients, then add or subtract the equations to remove that variable. For example, to solve 2x + 3y = 13 and 3x − y = 3, multiply the second by 3 to get 9x − 3y = 9, then add to the first to eliminate y. Always substitute the found value back into one original equation to find the second variable.

Worked example

Solve the simultaneous equations 2x + 3y = 13 and 4x − y = 5 using elimination.

1 $Multiply the second equation by 3 so the y-terms match: 12x - 3y = 15.$
2 $Add this to the first equation to eliminate y: 2x + 12x = 13 + 15, giving 14x = 28.$
3 $Solve for x: x = 2.$
4 $Substitute x = 2 into 4x - y = 5: 8 - y = 5, so y = 3.$

Answer: x = 2, y = 3

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Common mistake. Pupils multiply only one side of an equation when scaling it for elimination, breaking the equality. Always multiply every term on both sides of the equation by the same factor.

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