Year 10 Foundation · Simultaneous equations
Use one known value to find another
Learn how to solve a pair of simultaneous equations by graph, by elimination and by substitution, adjusting the equations first when you need to.
Definition
A pair of simultaneous equations is two equations that use the same two unknowns and are both true at the same time. One equation on its own has many possible answers: x + y = 10 works for 1 and 9, for 2 and 8, and for plenty more. The second equation pins the unknowns down to a single pair of values, and solving the pair means finding that one pair. You can find it from a graph, by combining the equations to remove one unknown, or by substituting one equation into the other.
Why Do I Have to Learn This?
Simultaneous equations solve two mysteries at once, like the price of two different snacks. They pop up whenever two things depend on each other.
Key Rules:
- 1. When one value is already known, substitute it into either equation and solve what is left to find the other unknown.
Remember:
- •Same signs subtract, opposite signs add; then substitute back so you finish with both values, not just one.
Don’t Forget...
- •Adding the equations when the matching terms have the same sign. 2y + 2y makes 4y, so nothing is eliminated; terms with the same sign need a subtraction, and only opposite signs cancel when you add.
- •Multiplying only some of the terms when scaling an equation. The equation stays true only if every term on both sides is multiplied, including the number on the right-hand side.
- •Stopping after finding the first unknown. The solution is a pair of values, so substitute the value you found back into an equation to find the second, then check both in the equation you have not used yet.
Worked Example:
Solve the simultaneous equations 3x + 2y = 16 and x + 2y = 8.
Both equations contain 2y with the same sign, so subtract the second from the first: 3x - x = 2x and 16 - 8 = 8, giving 2x = 8.
Solve for x: x = 4.
Substitute x = 4 into x + 2y = 8: 4 + 2y = 8, so 2y = 4 and y = 2.
Check the pair in the other equation: 3 x 4 + 2 x 2 = 12 + 4 = 16. Correct.
Answer: x = 4 and y = 2
Why It Works:
Both equations are true for the same pair of values, so adding them, subtracting them, or multiplying one through by a number produces another equation that the pair still satisfies. Arranging this so one unknown cancels leaves a true equation in a single unknown, which ordinary solving handles. The graph says the same thing: each line shows every pair that fits one equation, so the crossing point is the only pair that fits both.
Another Example:
Solve the simultaneous equations 4x + 3y = 23 and 2x - y = 9.
No terms match yet, so multiply the whole second equation by 3: 6x - 3y = 27.
The y terms are now 3y and -3y, opposite signs, so add the equations: 10x = 50, giving x = 5.
Substitute x = 5 into 2x - y = 9: 10 - y = 9, so y = 1.
Check the pair in the first equation: 4 x 5 + 3 x 1 = 20 + 3 = 23. Correct.
Answer: x = 5 and y = 1
Animated Example:
Free Use one known value to find another Worksheet
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Before this: Equations, inequalities and formulae (Year 9) · Next: Equations and formulae (Year 11 Foundation)
Let’s Practise the Concept
Step 1 of 3
Put them in order
Tap the numbers from smallest to largest.
In order so far: nothing yet