Study case: solving an algebraic fraction equation In this GCSE Algebraic Fractions Equation Problem Tutor, I use one clear study case to show how algebraic fractions can be solved without panic. The problem is: solve 3/(x + 1) = 2/(x - 2). The final answer is x = 8. I like this example because it tests a very important idea: denominators can be removed by multiplying through, but students must still check that the answer does not make a denominator zero. Before solving, I ask the student to identify the restrictions. In this equation, x cannot be -1 because x + 1 would be zero, and x cannot be 2 because x - 2 would be zero. These values are not allowed. This step is important because algebraic fractions are not only about solving; they are also about understanding where the expression is defined. Method 1 is cross multiplication. Since 3/(x + 1) = 2/(x - 2), I multiply diagonally to get 3(x - 2) = 2(x + 1). Expanding gives 3x - 6 = 2x + 2. Subtracting 2x from both sides gives x - 6 = 2. Adding 6 gives x = 8. I then check that x = 8 is allowed, and it is not one of the restricted values. So the solution is x = 8. Method 2 is multiplying by the common denominator. The common denominator is (x + 1)(x - 2). I multiply every term in the equation by this denominator. On the left, the x + 1 cancels, leaving 3(x - 2). On the right, the x - 2 cancels, leaving 2(x + 1). This gives the same equation as Method 1. I like this method because it shows why cross multiplication works and helps students avoid using it blindly. Method 3 is substitution checking. After finding x = 8, I substitute it back into the original equation. The left side is 3/(8 + 1) = 3/9 = 1/3. The right side is 2/(8 - 2) = 2/6 = 1/3. Both sides match, so the answer is correct. This check is useful because algebraic fraction equations can produce errors if brackets are expanded incorrectly or if a restricted value is accidentally included. After the three methods, I compare them with the student. Cross multiplication is quick. Multiplying by the common denominator is more explanatory. Substitution checking proves the result in the original equation. I want students to have all three routes because algebraic fractions often become more complex in higher GCSE questions. A student who understands the denominator structure is much more confident when the fractions have larger expressions. The most common mistakes are forgetting brackets after cross multiplying, multiplying only one side of the equation, losing a negative sign, cancelling terms incorrectly, and not checking restrictions. I use these mistakes as teaching clues. If a student writes 3x - 2 instead of 3(x - 2), I return to bracket expansion. If they cancel across addition, I review the difference between factors and terms. This topic links to simplifying algebraic fractions, solving rational equations and rearranging formulae. It also prepares students for A-Level algebra where fractions with expressions become more common. I teach students to slow down, write the common denominator clearly and keep equal signs aligned. Clear working prevents many avoidable errors. A useful extension is to solve 1/(x - 1) + 2/(x + 1) = 1. This requires combining fractions or multiplying through by a common denominator. Another extension is to create an equation where one solution is not allowed because it makes a denominator zero. These examples help students understand why checking restrictions matters. In one-to-one tutoring, I would finish by asking the student to solve a similar problem and explain each cancellation. I want the student to say which factor cancels and why. That spoken explanation shows genuine understanding rather than memorised cross multiplication. For support with this topic, students can book one-to-one tutor services: https://www.mastermathstutoring.co.uk/services. For the wider course page, visit GCSE and IGCSE Maths tutoring: https://www.mastermathstutoring.co.uk/gcse-igcse. To ask about lessons or availability, use contact MasterMaths Tutoring: https://www.mastermathstutoring.co.uk/contact-me.