Elimination by addition
Objective. Eliminate one unknown by adding equations.
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Learn: the key idea
Elimination by addition solves a pair of simultaneous equations by adding them together to cancel one variable, which works when that variable has opposite-signed coefficients in the two equations. First check that one variable's coefficients are equal in size but opposite in sign; if not, multiply one or both equations so they match. Adding the equations then leaves a single equation in one variable, which is solved and substituted back to find the other. For example, with 3x + y = 10 and 2x − y = 5, adding gives 5x = 15, so x = 3, and substituting gives y = 1.
Worked example
Solve the simultaneous equations 4x + y = 14 and 2x − y = 4 by elimination.
- 1The y terms have opposite signs (+y and −y), so add the equations.
- 2Adding: (4x + 2x) + (y − y) = 14 + 4, giving 6x = 18.
- 3Divide by 6: x = 3.
- 4Substitute x = 3 into 4x + y = 14: 12 + y = 14, so y = 2.
Answer: x = 3, y = 2
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Start practisingCommon mistake. Pupils sometimes add equations where the matching variable has the same sign rather than opposite signs, which doubles that variable instead of eliminating it. Check the signs are opposite before choosing to add rather than subtract.
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