IGCSE Maths / Simultaneous equations
Setting up simultaneous equations from a problem
Edexcel International GCSE Maths (4MA1), chapter 7: Simultaneous equations, specification point 2.6b-f. Free lesson.
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A problem with two unknowns needs two separate facts, which you turn into a pair of simultaneous equations. Solve for both unknowns and state what each value means.
How to set up the equations
One equation is not enough to find two unknowns. You need two independent facts, meaning two separate pieces of information about the unknowns. Give each unknown a letter and say what it represents, including its units. Then turn each fact into an equation. Two unknowns need two separate facts and two equations. Writing the pair of equations clearly can earn marks before you start solving.
How to solve the apple and banana problem
Let a be the cost of one apple in pence and b the cost of one banana in pence. Using pence keeps the numbers whole. The first sentence gives the first equation, and the second sentence gives the second.
Keep the units consistent
You can set up the equations in pounds or pence. Either works if both equations use the same unit. Pence avoids decimals, so it can reduce the chance of an arithmetic slip. Mixing pounds in one equation with pence in the other gives an incorrect result. Use one unit throughout the pair of equations. For example, in pence the apple and banana equations are 2a + 3b = 160 and 4a + b = 140.
Check that you have answered the whole question
If a problem asks for the cost of each item, finding just one cost is not enough. Substitute back to find the other unknown too, then give both values with their meanings and units. Two unknowns mean two values in the final answer. A correct value for a alone does not answer the question about both items.
Try the pen and ruler problem
Let p be the cost of one pen in pence and r the cost of one ruler in pence. Write one equation for each sentence, then solve for both costs.