Year 9 · Simultaneous equations
Use one known value to find another
Learn how to solve a pair of simultaneous equations by drawing their graphs, by adding or subtracting them, and by substituting one equation into the other.
Definition
A pair of simultaneous equations is two equations that use the same two unknowns and are both true at the same time. On its own, an equation like x + y = 10 has endless possible answers, but pair it with x - y = 2 and only one pair of values fits both. Solving the pair means finding that single pair of values. Each equation draws a straight line, and the solution is the point where the two lines cross.
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. If you already know the value of one unknown, substitute it into either equation and solve to find the other.
Remember:
- •Make one unknown disappear, solve for the one that is left, then substitute back to find the other.
Don’t Forget...
- •Adding the equations when the matching terms have the same sign, so nothing cancels. With +2y in both equations, adding gives 4y and the unknown survives; terms only disappear when subtracting matching signs or adding opposite signs makes them sum to zero.
- •Multiplying only some of the terms when adjusting an equation, for example turning x + 2y = 8 into 3x + 2y = 24. An equation stays true only if every term on both sides is multiplied by the same number, so the correct result is 3x + 6y = 24.
- •Stopping after finding one unknown and giving a single value as the answer. The solution is a pair of values: substitute the value you found back into an equation to find the other unknown, then check both in the equation you have not used.
Worked Example:
Solve the simultaneous equations 3x + 2y = 12 and x + 2y = 8.
Both equations contain +2y, so subtract the second from the first to eliminate y: (3x + 2y) - (x + 2y) = 12 - 8.
This leaves 2x = 4, so x = 2.
Substitute x = 2 into x + 2y = 8: that gives 2 + 2y = 8, so 2y = 6 and y = 3.
Check the pair in the other equation: 3 x 2 + 2 x 3 = 6 + 6 = 12, which is correct.
Answer: x = 2 and y = 3
Why It Works:
Both equations state true facts about the same x and y, and subtracting one true equation from another, or scaling every term of one by the same number, produces another equation that is still true for those values. Choosing the step so that one unknown cancels leaves a single equation in one unknown, which you already know how to solve.
Another Example:
Solve the simultaneous equations y = 2x + 1 and 3x + y = 16.
The first equation gives y by itself, so replace y in the second equation with 2x + 1: 3x + (2x + 1) = 16.
Simplify: 5x + 1 = 16, so 5x = 15 and x = 3.
Substitute x = 3 back into y = 2x + 1: y = 2 x 3 + 1 = 7.
Check in the second equation: 3 x 3 + 7 = 16, which is correct.
Answer: x = 3 and y = 7
Animated Example:
Free Use one known value to find another Worksheet
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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