Rearranging Complex Formulae

The Challenge: Isolating a Repeated Subject

Let's say you're given the formula 5x = 2x + 9 and asked to solve for x. You know exactly what to do:

1. Collect the x terms: 5x - 2x = 9
2. Simplify: 3x = 9
3. Isolate x: x = 3

But what if you're asked to make x the subject of the formula ax = bx + c? If you try the same approach, you get ax - bx = c. You've collected the terms, but you're still left with two instances of x. You can't combine ax and bx into a single term without knowing what a and b are.

The key insight is this: to turn multiple terms containing your subject into a single instance, you must use factorization.

ax - bx = c becomes x(a - b) = c.

Now, x appears only once. The final step is simple: divide by the bracket.

x = c / (a - b)

This core idea—collect, factorize, isolate—is the foundation of the method we'll master today.

A Four-Step Algorithm for Success

The most complex-looking rearrangement problems can be broken down into a predictable, four-step algorithm. Memorising and practising this process will transform these questions from a challenge into a guaranteed source of marks.

The following video from Cognito demonstrates this four-step method very clearly. As you watch, focus on how each step moves you closer to isolating the subject.

To summarise and formalise the process from the video, here is your algorithm:

1. Clear Fractions and Expand Brackets: Your first goal is to get a "flat" line of individual terms being added or subtracted.
   • If there are fractions, multiply every term by the denominator(s) to eliminate them.
   • If there are brackets, expand them to release the terms inside.
2. Collect Subject Terms: Move all terms containing the variable you want to make the subject to one side of the equals sign. Move all other terms to the opposite side.
3. Factorize: Take the subject variable out as a common factor from all the terms you collected on one side. This is the crucial step that leaves you with a single instance of your subject.
4. Isolate the Subject: Divide both sides by the bracket you created during factorization. This leaves the subject by itself.

Applying the Method: Worked Example

Let's apply this method to a typical exam-style question.

Problem: Make b the subject of the formula c = (a - b) / (2b - 5).

• Step 1: Clear Fractions
  The denominator is (2b - 5). We multiply both sides by this to clear the fraction.
  $c(2b-5)=a-b$
  Now we have brackets, so we expand them.
  $2bc-5c=a-b$

• Step 2: Collect Subject Terms
  The subject is b. We need to get all terms with b on one side (let's choose the left) and everything else on the other.
  The -b on the right needs to move left (becomes +b).
  The -5c on the left needs to move right (becomes +5c).
  $2bc+b=a+5c$

• Step 3: Factorize
  The left side has two terms containing b. We factor b out.
  $b(2c+1)=a+5c$
  Watch out for the common error here! b divided by b is 1, not 0.

• Step 4: Isolate the Subject
  b is being multiplied by the bracket (2c + 1). The inverse operation is division.
  $b=\frac{a+5c}{2c+1}$
  And we are done. The formula is rearranged with b as the subject.

Building Your Pattern Recognition

To improve your speed and confidence, you need to see this pattern in different contexts. The next video provides more examples, including some 'try it yourself' moments.

Time for Practice

Your goal is to become so familiar with the four-step method that it becomes automatic. Use the following problems to train this skill. For each one, methodically apply the steps: Clear/Expand -> Collect -> Factorize -> Isolate.

Avoiding Common Errors

You've mentioned that you're prone to small calculation errors. In these questions, marks are often lost not because of a lack of understanding, but due to simple mistakes in the process. Let's proactively address them.

The key takeaways from that reading for you are:

• Be precise with negatives: When you move a term across the equals sign, meticulously change its sign.
• Operations apply to everything: When you multiply or divide one side of an equation, you must multiply or divide every term on that side.
• Factorization is not optional: As the article highlights, if the subject (π in their example) appears twice, you must factorize to isolate it. There is no other way. This confirms that our four-step method is the correct and most efficient strategy.

Conclusion

You now have a powerful and reliable algorithm for one of the more challenging algebraic manipulation tasks at IGCSE. By turning it into a systematic process, you remove the guesswork and can confidently secure the marks on these questions.

Key Takeaways:

• When a formula has the subject variable in multiple terms, your main strategy is to collect, factorize, and isolate.
• The four-step algorithm provides a clear path to the solution:
  1. Clear/Expand: Get a flat line of terms.
  2. Collect: Get all subject terms on one side, everything else on the other.
  3. Factorize: Take out the subject as a common factor.
  4. Isolate: Divide by the resulting bracket.
• Practice this sequence until it becomes second nature to avoid small errors under pressure.

Preview of the Next Lesson:
In our next lesson, we will move into a new topic: simultaneous equations. We'll start by learning the methods of elimination and substitution to solve pairs of linear equations, a fundamental skill for solving problems where two different conditions must be met at the same time.