Simultaneous Equations
This topic covers 5 learning steps, guiding your child from the basics through to confident problem-solving. Each step includes a worked example and adaptive practice questions.
What Your Child Will Learn
- Substitution method intro
Solve by substitution when one variable is given - Elimination method
Solve by adding/subtracting equations - Word problems to equations
Form simultaneous equations from words - Mixed methods
Choose the best method to solve - Challenge problems
Multi-step simultaneous equation problems
Before This Topic
Your child should be comfortable with:
- Two-Step Equations (Year 6)
- Expanding & Factorising (Year 8)
Common Mistakes
- Solving for one variable but forgetting to find the other (e.g. finding x = 3 but not substituting back to find y)
Simultaneous equations always have TWO unknowns, so you always need TWO answers. Once you find one variable, substitute it back into either original equation to find the other. Always write both values in your answer. - Subtracting equations when they should be added (or vice versa) during elimination
Look at the signs of the terms you want to eliminate. If they are the same sign, SUBTRACT the equations. If they are different signs, ADD them. Same signs subtract, different signs add.
Tips for Parents
- Start with a real-life puzzle: "2 coffees and 1 tea cost £7. 1 coffee and 1 tea cost £4.50. How much is a coffee?" Subtracting gives: 1 coffee = £2.50, so 1 tea = £2.
- Practise substitution first — it is more intuitive. If y = x + 3 and 2x + y = 12, replace y with (x + 3) to get 2x + x + 3 = 12, so 3x = 9, x = 3.
- For elimination, line up the equations vertically and look for matching coefficients. If they do not match, multiply one or both equations to create a match.
- Always check the answer by substituting both values back into BOTH original equations. If both work, the answer is correct.
Key Words
- Simultaneous equations — Two equations with two unknowns that are both true at the same time — you find the values that satisfy both.
- Substitution — Replacing one variable with an expression — if y = x + 3, replace y in the other equation with (x + 3).
- Elimination — Adding or subtracting equations to remove one variable, leaving an equation with just one unknown.
- Coefficient — The number in front of a letter — in 3x + 2y = 10, the coefficient of x is 3 and the coefficient of y is 2.
- Solution — The pair of values (x, y) that makes both equations true at the same time.
Where This Fits
Before this topic: Children should be confident solving two-step linear equations and understand substitution into expressions.
After this topic: Simultaneous equations lead to graphical solutions (where two lines intersect), non-linear simultaneous equations at GCSE, and real-world modelling problems.
How MathCraft Teaches This
In MathCraft, Simultaneous Equations is taught through the Algebra & Arithmetic adventure track. Your child follows guided lessons with friendly characters, works through examples step by step, then practises with questions that adapt to their level.
The adaptive engine tracks mastery across all 5 steps, revisiting concepts your child finds tricky and advancing when they're ready. Parents can see detailed progress in the Parent Dashboard.
Practise Simultaneous Equations with MathCraft
Step-by-step lessons, worked examples, and adaptive practice — all wrapped in an adventure game your child will love.
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