Simultaneous equations practice questions and worked solutions

Substitution is faster when one variable is already isolated, or can be isolated in one step without introducing fractions. If an equation reads y = 2x + 1, substituting that expression directly into the other equation is almost always the shortest route to an answer.

Elimination is faster when the equations are both in the form ax + by = c, particularly when adding or subtracting them cancels a variable immediately. A pair such as x + y = 10 and x − y = 4 is built for elimination, and a student who sets up substitution instead has made more work than the question required.

Neither method is more correct, and marks are awarded for either. What examiners are looking for is a correct and clearly set-out approach, so spending three seconds deciding which is cleaner is a better investment than starting immediately with whichever is more familiar.

Substitute the values back into the equation you did not use when finding the second variable. Checking against the equation you just used will confirm your arithmetic in that step but cannot catch an error made earlier, so it gives false confidence in exactly the situation where a check matters.

For word problems, the check has a second part: confirm that the answer makes sense in context. A negative number of tickets or a fractional number of buses signals a set-up error rather than an arithmetic one, and the marks lost there are for translating the problem rather than for solving it.

When the variables cancel and leave a false statement such as 0 = 6, the lines are parallel and distinct, so there is no solution. When they cancel and leave a true statement such as 0 = 0, the two equations describe the same line and every point on it is a solution.

These outcomes feel like something has gone wrong, and students often assume they have made a mistake and start over. Recognising the two signatures saves time and picks up marks, because questions asking you to determine the number of solutions are testing precisely this recognition rather than the algebra.

There is a parameter version of the same question at senior level, asking for the value of a constant that makes a system have no solution. Those are answered by setting the coefficients proportional to each other, which follows directly from the lines being parallel.

Most marks in this topic are lost before any algebra happens. Define the variables explicitly in words, write the two relationships as separate sentences, and only then translate each sentence into an equation. Attempting to write the equations straight from the paragraph is where the translation errors creep in.

Watch for relationships expressed as differences and comparisons, which reverse easily. A statement that one number is four more than another becomes a minus sign on one side or a plus on the other depending on which variable you subtract from which, and getting it backwards produces a clean-looking answer that is wrong.