IGCSE Maths / Simultaneous equations
Simultaneous equations by inspection revision notes
Edexcel International GCSE Maths (4MA1), chapter 7: Simultaneous equations, specification point 2.6a-f. Free lesson.
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Solving by inspection means spotting the values of the unknowns when they are easy to see. The values must satisfy both equations, and you should switch to elimination or substitution when the answers are not obvious.
When inspection works
Inspection works when an equation gives you a value directly, or when the answers are small whole numbers that are easy to spot. A solution to simultaneous equations must make both equations true at the same time. If one equation contains just one unknown, you can read off its value straight away. For two equations such as x + y = 7 and x − y = 1, look for numbers that both add to 7 and differ by 1.
Use a value given directly
Once one unknown is known, substitute it into the other equation. This leaves an equation with one unknown, which you can solve.
Check both equations
A pair that works in only one equation is not a solution. Substitute both values into each equation to check that both are true. State both unknowns, because one correct value alone does not give the full answer. Check the equation you used to find the values and the other equation too. For x + y = 7 and x − y = 1, x = 5 and y = 2 fits the first equation because 5 + 2 = 7. But it fails the second because 5 − 2 = 3, not 1.
Know when to change method
Inspection is useful when the answer is obvious, but it can become guesswork when the numbers are large or fractional. Use elimination or substitution instead when you cannot see a small whole-number pair. For x + y = 12 and x − y = 2, the small whole numbers are easy to spot: 7 and 5. Check: 7 + 5 = 12 and 7 − 5 = 2.