Form simultaneous equations

Objective. Translate paired constraints into two equations.

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2. Worked example
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Learn: the key idea

Forming simultaneous equations means translating a worded problem with two linked unknowns into a pair of algebraic equations. Each sentence of information usually becomes one equation, with consistent letters chosen to represent each unknown quantity throughout. For example, if a adult ticket costs x pounds and a child ticket costs y pounds, the statement '2 adults and 3 children cost £29' becomes 2x + 3y = 29. Once both equations are formed from the two pieces of given information, they can be solved using elimination or substitution to find the value of each unknown.

Worked example

Two numbers have a sum of 18 and a difference of 4. Form a pair of simultaneous equations to represent this and find the two numbers.

1 $Let the numbers be x and y, giving the equations x + y = 18 and x - y = 4.$
2 $Add the two equations to eliminate y: 2x = 22.$
3 $Divide by 2: x = 11.$
4 $Substitute into x + y = 18: 11 + y = 18, so y = 7.$

Answer: The numbers are 11 and 7

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Common mistake. Pupils sometimes mix up which letter represents which unknown between the two equations, such as swapping x and y in the second equation, creating an inconsistent pair. Define each letter clearly at the start and keep it consistent in both equations.

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