SecondaryMaths

Simultaneous equations – When to add and when to subtract

When using the elimination method to solve simultaneous equations, students can often be unsure whether to add or subtract, notes Colin Foster

by Colin Foster
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SecondaryMaths

The difficulty

Look at the four pairs of simultaneous equations below:

Which ones could you solve by adding the equations together? Which ones could you solve by subtracting one equation from the other? Students may be unsure and not know how to decide.

The solution

Don’t worry about solving the equations just yet. All I want you to do is simply add together each pair of equations. And also subtract each pair of equations. See what you get.

By adding, students should obtain the following:

And by subtracting the second equation from the first equation:

Students may not bother to write the 0𝑥 and 0𝑦 where there are no 𝑥 and 𝑦 terms, and this is fine.

They may make errors, particularly when subtracting the negative terms – so for the subtractions, they may end up with the wrong answers shown in red below:

Writing out the difficult subtractions explicitly may help:

When does adding eliminate an unknown?
This happens when two terms are equal in magnitude, but of opposite sign
(e.g., 2𝑦 and –2𝑦).

When does subtracting eliminate an unknown?
This happens when two terms are equal in magnitude, and of the same sign
(e.g., –3𝑦 and –3𝑦).

Sometimes, it can help if students remember the following: When the Signs are the Same you Subtract.

Can you find a pair of equations where either adding or subtracting will lead to elimination of one of the unknowns?

An example would be 3𝑥 + 2𝑦 = 11 and 3𝑥 – 2𝑦 = 7.

The solution to all of these pairs of equations is 𝑥 = 3, 𝑦 = 1.

Checking for understanding

To assess students’ understanding, ask them to create four pairs of simultaneous equations of their own, two of which can be solved by adding the equations, and two of which can be solved by subtracting the equations.

They should label clearly which are which.

Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University and has written numerous books and articles for mathematics teachers; for more information, visit foster77.co.uk