Mathematics · Algebra and simultaneous equations
Substitution or elimination for simultaneous equations? I always pick the slow one.
Look at the coefficients before you choose. There is a quick test, and it takes about three seconds.
The test
Use elimination when a variable has the same or easily matched coefficients. If you can multiply one equation by a small whole number and make a pair of coefficients match, elimination is faster and makes fewer fractions.
Use substitution when one equation already has a variable on its own, or with coefficient 1, so rearranging costs nothing.
Always use substitution when one equation is non-linear — a quadratic, a circle, anything with a squared term. Elimination does not work on those, and this is the case most people get stuck on.
Elimination, worked
3x + 2y = 16
5x - 2y = 8The y coefficients are already +2 and −2. Add the equations and y vanishes:
8x = 24, so x = 3Substitute back into the first: 9 + 2y = 16, so y = 3.5.
Had they been +2 and +4, multiplying the first by 2 would match them, then subtract. Matching signs subtract; opposite signs add. Getting this backwards is the most common arithmetic slip in the topic, and writing "adding" or "subtracting" next to your working makes it visible.
Substitution, worked
y = 2x - 1
x^2 + y^2 = 25Elimination is useless here. Substitute the first into the second:
x^2 + (2x - 1)^2 = 25
x^2 + 4x^2 - 4x + 1 = 25
5x^2 - 4x - 24 = 0Solve the quadratic, then find each y from y = 2x − 1.
Two things to be careful about. Square the whole bracket, including the cross term — (2x − 1)² is not 4x² + 1. And having found two x values, find the matching y for each, giving two coordinate pairs. Stopping at the x values loses half the marks, and this is where most of them go.
Checking
Substitute both answers into the equation you did not use for the final step. If they satisfy it, you are right. It takes fifteen seconds and catches sign errors that are otherwise invisible, because a wrong answer still looks tidy.
For a geometric question — a line meeting a circle — the number of solutions is itself meaningful. Two solutions means the line cuts the circle twice, one means it is a tangent, and none means it misses. If a question asks you to show a line is a tangent, you are being asked to show the resulting quadratic has exactly one root, which means its discriminant is zero.
Working from a past paper? Mark your attempt against the published mark scheme before reading an answer like this one. Finding out where you lost the marks is worth more than seeing a correct solution, because the correct solution rarely tells you what you did instead.
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