GCSE Maths · Maths
Simultaneous equations: a clear study guide
Simultaneous equations describe two relationships that share the same unknowns. Solve them together, keep the algebra organised and check both values in both original equations.
What a simultaneous solution means
A pair of linear equations has a simultaneous solution when one pair of values makes both equations true. On a graph, the solution is the point where the two straight lines meet.
Write the equations clearly before choosing a method. Look for matching or easily matched coefficients when elimination is likely to be efficient.
- The solution is an ordered pair such as (x, y).
- Substitute the values into both original equations when checking.
- Keep negative signs attached to the term they belong to.
Elimination and substitution
For elimination, add or subtract the equations so that one variable disappears, then solve the remaining equation. If the coefficients do not match, multiply one or both equations first.
For substitution, rearrange one equation to make a variable the subject, substitute it into the other equation, and solve. Substitute the answer back to find the second variable.
Check the pair, not just the algebra
A quick check is to put both values into each original equation. In a word problem, explain what each value represents and reject a value that is impossible for the context, such as a negative length.
Quick check
What does the solution to two simultaneous equations represent on a graph?
Answer: The coordinates of the intersection of the two lines.
Why should both original equations be checked?
Answer: A pair is only correct if it satisfies both equations; checking also catches sign or substitution errors.
Common questions
Should I use elimination or substitution?
Choose the method that keeps the coefficients and arithmetic simplest. Elimination is often efficient when a variable already has matching coefficients; substitution is useful when one equation is already rearranged.
Can simultaneous equations have no solution?
Yes. Parallel lines have no intersection, so the equations do not share a solution. The question's algebra will show the inconsistency.