IGCSE Maths / Simultaneous equations
Simultaneous equations by substitution
Edexcel International GCSE Maths (4MA1), chapter 7: Simultaneous equations, specification point 2.6a. Free lesson.
Read the revision notes · IGCSE Maths past papers
Substitution solves two equations by making one letter the subject, then replacing that letter in the other equation. It is most useful when a letter already has a coefficient of 1 or −1, so the working stays simple.
When to choose substitution
Both equations must be true for the same pair of values. Substitution is usually the easier method when one letter has a coefficient of 1 or −1, because you can rearrange that equation without introducing awkward fractions. If neither equation has such a letter, elimination is usually more efficient. Substitution starts by rearranging an equation. Elimination starts by multiplying an equation so a pair of terms match. After the first move, both methods leave you with one equation in one unknown. Then you find the other unknown. Neither method earns more marks by itself. Choose the one…
How to substitute and solve
Make one letter the subject, so it is written in terms of the other letter. Replace it in the other equation, putting the replacement in brackets. This keeps the whole expression together when you multiply it.
Worked example: solve the pair
For x − 2y = 1 and 3x + 4y = 23, the x in the first equation has a coefficient of 1, so rearrange that equation to make x the subject. Then put the expression for x into the second equation.
Why the brackets matter
When you replace x with 2y + 1 in 3x, the 3 multiplies the whole expression, not just its first term. If you expand the bracket incorrectly, the later working no longer matches the original equation and the marks that follow can be lost. Correct: 3(2y + 1) = 6y + 3. Incorrect: 3(2y + 1) = 6y + 1. Keep the replacement in brackets until you expand every term.
Quick check and final routine
In y = 4x − 3 and 2x + y = 9, y is already on its own. Replace y in the second equation with 4x − 3, then solve and return to the original subject line for the other value. The solution is x = 2, y = 5. For the first worked example, the rearranged line is x = 2y + 1. Use this line to find x after finding y.