Rearranging Formulas Revision Notes for AQA GCSE Maths

Rearranging Formulas

Introduction

Rearranging formulas is a fundamental skill in algebra that involves changing the structure of an equation to make a specific variable the subject. When we rearrange a formula, we're essentially solving for one particular letter or variable, isolating it on one side of the equation. For example, if we have the equation $2x + z = 3(y + 2p)$ , we might want to rearrange it to make $y$ the subject, resulting in $y = \text{something}$ .

The key principle behind rearranging formulas is that we need to get the desired variable completely on its own on one side of the equation. This process is remarkably similar to solving equations, but with the added complexity that we're working with multiple variables rather than just numbers.

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The beauty of rearranging formulas is that it allows us to express any variable in terms of the others, making it incredibly useful for solving real-world problems where you know some values but need to find others.

The systematic approach to rearranging formulas

The most effective way to rearrange formulas is to follow a systematic seven-step method. This approach ensures you don't miss any important steps and helps you work through even complex rearrangements methodically.

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Critical Success Factor: Always work through the steps in order, but remember that you can skip steps that don't apply to your specific formula. Don't feel obligated to use every step for every problem.

The systematic method involves working through these key stages:

Step 1: Eliminate square root signs If your formula contains square root signs, remove them by squaring both sides of the equation. This simplifies the equation and makes it easier to work with.

Step 2: Remove fractions Clear any fractions by multiplying every term by the denominator. This step is crucial because fractions can make the subsequent steps much more complicated.

Step 3: Expand brackets Multiply out any brackets in the equation. This involves applying the distributive property to ensure all terms are fully expanded.

Step 4: Collect like terms Gather all terms containing the subject variable on one side of the equation and all other terms on the opposite side. This is where you start to isolate your target variable.

Step 5: Simplify to standard form Reduce the equation to the form $Ax = B$ , where $A$ and $B$ represent all the other variables and numbers. You may need to combine like terms or factor out common factors at this stage.

Step 6: Divide to isolate the variable Divide both sides by the coefficient of your subject variable to get the variable by itself.

Step 7: Handle squares if necessary If your subject variable is squared, take the square root of both sides to complete the rearrangement. Don't forget to include the $\pm$ symbol when taking square roots.

Special case: Subject appears in a fraction

When your subject variable appears in a fraction, you don't need to use every step of the method. You can skip the steps that don't apply to your particular situation.

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Worked Example: Making b the subject of a = (5b + 3)/4

Step 1: Skip (no square roots)

Step 2: Eliminate the fraction by multiplying every term by 4: $4a = 5b + 3$

Step 3: Skip (no brackets to expand)

Step 4: Collect subject terms on one side: $5b = 4a - 3$

Step 5: The equation is now in the form $Ab = B$ where $A = 5$ and $B = 4a - 3$

Step 6: Divide both sides by 5: $b = \frac{4a - 3}{5}$

Step 7: Skip (b isn't squared)

Final answer: $b = \frac{4a - 3}{5}$

Special case: Square or square root involved

When your subject variable appears as a square or under a square root, you'll need to use steps 1 and 7 appropriately.

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Worked Example: Making u the subject of v² = u² + 2as

Step 1: Skip (no initial square roots)

Step 2: Skip (no fractions)

Step 3: Skip (no brackets)

Step 4: Collect subject terms on one side: $u² = v² - 2as$

Step 5: Already in form $Au² = B$ where $A = 1$ and $B = v² - 2as$

Step 6: Skip (coefficient is 1)

Step 7: Take square root of both sides: $u = \pm\sqrt{v² - 2as}$

Final answer: $u = \pm\sqrt{v² - 2as}$

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Worked Example: Making n the subject of 2(m + 3) = √(n + 5)

Step 1: Square both sides to eliminate square root: $\[2(m + 3)]² = n + 5$

Step 2: Skip (no fractions)

Step 3: Expand brackets: $4(m² + 6m + 9) = n + 5$ $4m² + 24m + 36 = n + 5$

Step 4: Collect subject terms: $n = 4m² + 24m + 36 - 5$ $n = 4m² + 24m + 31$

Final answer: $n = 4m² + 24m + 31$

Special case: Subject appears twice

Sometimes your subject variable appears in multiple places in the formula. When this happens, you'll need to use factoring techniques, usually in step 5.

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Worked Example: Making p the subject of q = (p + 1)/(p - 1)

Step 1: Skip (no square roots)

Step 2: Clear the fraction by multiplying both sides by $(p - 1)$ : $q(p - 1) = p + 1$

Step 3: Expand brackets: $qp - q = p + 1$

Step 4: Collect subject terms on one side: $qp - p = q + 1$

Step 5: Factor out p (since it appears twice): $p(q - 1) = q + 1$

Step 6: Divide both sides by $(q - 1)$ : $p = \frac{q + 1}{q - 1}$

Final answer: $p = \frac{q + 1}{q - 1}$

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Key Insight for Repeated Subjects: When your subject variable appears multiple times, factoring is essential. Look for opportunities to extract the subject as a common factor, which allows you to isolate it effectively.

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Key Points to Remember:

Rearranging formulas follows a systematic seven-step method that's similar to solving equations
You don't always need to use every step - skip the ones that don't apply to your specific formula
When the subject appears in a fraction, focus on clearing the fraction early in the process
Square roots and squares require special attention in steps 1 and 7
If the subject appears twice, you'll need to factor it out to isolate it properly
Always work systematically and don't rush through the steps
Practice with different types of formulas to build confidence with the method

Rearranging Formulas (AQA GCSE Maths): Revision Notes