GCSE

Simultaneous Equations

Two equations, two unknowns — this might look intimidating, but the method is actually very logical once you see it in action.

What Your Child Is Learning

A simultaneous equation problem gives your child two equations with two unknown values (usually \(x\) and \(y\)). The goal is to find the values that make both equations true at the same time.

Here is a simple example:

$$3x + y = 10$$ $$x + y = 4$$

There is exactly one pair of values (\(x = 3\), \(y = 1\)) that works in both equations simultaneously. That is what your child is learning to find.

This comes up everywhere in real life — any time you have two conditions that must both be satisfied. It is a genuinely useful skill.

How Schools Teach It Now

Schools teach two main methods. Your child will usually learn both and choose the one that suits the question:

Elimination (the most common method at GCSE):

  1. Line up the two equations.
  2. Make the coefficients (the numbers in front) of one variable the same in both equations.
  3. Add or subtract the equations to eliminate that variable.
  4. Solve for the remaining variable, then substitute back to find the other.

Substitution (useful when one equation is already simple):

  1. Rearrange one equation to express one variable in terms of the other (e.g. \(y = 10 - 3x\)).
  2. Plug that expression into the second equation.
  3. Solve, then substitute back.
Parent tip
If your child finds elimination confusing, ask them to think of it as a "balancing act" — both sides of each equation are balanced, so you can add or subtract whole equations from each other without breaking the balance.

Common Mistakes to Watch For

Classic mistake
Subtracting when they should add (or vice versa). The rule is: if the matching coefficients have the same sign, subtract the equations. If they have different signs, add them. Students often get this the wrong way round, and the variable does not cancel out.
Watch out
Sign errors when eliminating. Subtracting a negative number is where most mistakes happen. If the second equation has \(-2y\) and they are subtracting, that \(-2y\) becomes \(+2y\). This catches a lot of students out.
Easy marks lost
Not checking the answer in both equations. Many students find \(x\) and \(y\), write them down, and move on. Checking catches slips — substitute the values back into both original equations and confirm they work.

Try It Together

Here is one to work through together:

$$3x + y = 10 \quad \text{...(1)}$$ $$x + y = 4 \quad \text{...(2)}$$

Step by step using elimination:

  1. Both equations have \(1y\) — the coefficients already match.
  2. Same sign on \(y\), so subtract equation (2) from equation (1):
    $$(3x + y) - (x + y) = 10 - 4$$$$2x = 6$$$$x = 3$$
  3. Substitute \(x = 3\) back into equation (2): \(3 + y = 4\), so \(y = 1\).
  4. Check in equation (1): \(3(3) + 1 = 9 + 1 = 10\). Correct.
Conversation starter
Ask your child: "If I told you that two numbers add up to 10 and the difference between them is 4, what are they?" (7 and 3.) That is a simultaneous equation in disguise — they probably solved it without even realising.

Let Them Practise

Like most algebra topics, simultaneous equations become easier with practice. Once the method clicks, students can solve these quickly and reliably.

We have a free interactive tool where your child can practise solving simultaneous equations step by step:

Simultaneous Solver

Free interactive practice — work through each step with instant feedback.

Practise Now on MaffsGames

MaffsGames needs no account and no sign-up; what it records while your child plays is set out on the privacy page.

All parent guides · MaffsGames home