For a pair of linear simultaneous equations, use elimination when both equations are written in ax + by = c form with no variable conveniently alone — scale one or both equations so a variable's coefficients match, then add or subtract. Use substitution when one equation already has a variable isolated (e.g. "y = ..."), or is trivially easy to rearrange into that form. Both methods always give the correct answer; the choice is about speed and avoiding unnecessary fractions.
Both methods always work — the difference is speed and error risk, not correctness. Both methods also rely on the same skill of solving linear equations once a single variable has been isolated. Picking the wrong method doesn't cost you the answer, but it costs you time and introduces more chances for arithmetic slips, especially with fractions. Picking correctly turns a 5-line solution into a 3-line one.
Use substitution if...
One variable is already isolated (or trivially easy to isolate) in one equation.
Use elimination if...
Both equations are in ax + by = c form, with no variable conveniently alone.
Elimination works by adding or subtracting the two equations to cancel one variable entirely.
Example — solve 3x + 2y = 16 and 5x − 2y = 8
(3x + 5x) + (2y − 2y) = 16 + 88x = 24x = 33(3) + 2y = 169 + 2y = 16y = 3.5Same sign coefficients → subtract. Opposite sign coefficients → add. If the coefficients of the variable you're eliminating have the same sign (both +2 or both −2), subtract one equation from the other. If they have opposite signs (+2 and −2), add the equations. This one rule covers every elimination case.
Elimination still works even when the coefficients aren't equal — you just need to scale one or both equations first.
Example — solve 2x + 3y = 13 and 5x + 2y = 13
(2x + 3y = 13) × 2 → 4x + 6y = 26(5x + 2y = 13) × 3 → 15x + 6y = 39(15x − 4x) + (6y − 6y) = 39 − 2611x = 13x = 13/11Substitution is often the faster choice when one equation is already written as "y = ..." or "x = ...", or can be rearranged into that form in one step.
Example — solve y = 2x − 1 and 3x + 4y = 17
3x + 4(2x − 1) = 173x + 8x − 4 = 1711x = 21x = 21/11Recognise this pattern: whenever one equation is already written as "y = ..." or "x = ...", substitution is almost always faster than rearranging into ax + by = c form and eliminating. Don't force elimination just because it's the "standard" method — read what the question already gives you.
Example — solve x = y + 3 and 2x + 5y = 13
2(y + 3) + 5y = 132y + 6 + 5y = 137y = 7y = 1x = 1 + 3 = 4Common mistake with substitution
When substituting an expression like (y + 3) or (2x − 1) into the second equation, students often forget to put brackets around it — writing 2y + 3 + 5y = 13 instead of 2(y + 3) + 5y = 13. Always wrap the substituted expression in brackets, then expand carefully.
The method (substitution or elimination) is only half the challenge in O-Level word problems. Setting up correct equations from the words is where most marks are actually lost.
Example — 3 apples and 2 oranges cost $3.10. 5 apples and 4 oranges cost $5.60. Find the cost of one apple and one orange.
3a + 2o = 3.105a + 4o = 5.606a + 4o = 6.20(6a − 5a) + (4o − 4o) = 6.20 − 5.60a = 0.603(0.60) + 2o = 3.101.80 + 2o = 3.10o = 0.65Always define your variables first
Writing "let a = apple" without specifying the unit (cost in dollars? weight in grams?) is a common way to lose a mark in word problems. State exactly what each variable represents before writing the equations.
Can I always use substitution, even when no variable is isolated?
Yes — substitution always works mathematically. But when neither equation has an isolated variable, substitution requires an extra rearranging step first (dividing to make a variable the subject), which elimination skips entirely. For two equations already in standard ax + by = c form, elimination is usually 1–2 lines shorter.
What if I eliminate the wrong variable — do I have to start over?
No — eliminating either variable first is equally valid. If eliminating y looks messier than eliminating x (or vice versa) after scaling, you're free to switch which variable you target. Choose whichever produces smaller, simpler numbers to scale by.
My answer has a decimal or fraction — did I make a mistake?
Not necessarily. Many O-Level simultaneous equation pairs are deliberately constructed to give whole-number answers, but not all of them are — particularly in word problems involving money or measurements. Double-check your working with substitution back into both original equations rather than assuming a non-integer answer is wrong.
How do I check my final answer is correct?
Substitute both values back into the original equation you did not use for your final substitution step. If both sides balance, your answer is correct. This check takes under a minute and catches nearly every arithmetic slip before you move to the next question.
Are simultaneous equations in O-Level E-Math always linear?
Yes — under the Algebra topic, simultaneous equations are always a pair of linear equations, solved using substitution or elimination as covered in this post. A separate topic later in the syllabus, under Graphs, covers finding the intersection of a line and a curve (such as a quadratic) using a graphical approach — that's a distinct skill from this one and uses a different method.
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