Welcome to Rearranging Formulae!

Have you ever solved an equation like \(2x + 3 = 11\)? If so, you already have the superpowers needed for this chapter! Rearranging a formula is just like solving an equation, but instead of finding a single number at the end, you rearrange the letters so that a different letter stands alone.

In this guide, we will break down what formulae are, what it means to change the subject, and how to use inverse operations step-by-step so you can tackle any problem with confidence.

1. The Basics: What is a Formula and What is the "Subject"?

Let's start by defining two key terms you will see all the time in algebra:

Formula (plural: formulae or formulas): A mathematical rule or relationship connecting two or more variables, written with an equals sign (\(=\)).
Example: \(C = 2\pi r\) (the formula for the circumference of a circle) or \(s = \frac{d}{t}\) (speed equals distance divided by time).

Subject of a Formula: The single variable that is completely on its own on one side of the equals sign, with a positive coefficient of \(+1\).
Example: In the formula \(y = mx + c\), the letter \(y\) is the subject because it sits alone on the left-hand side with no numbers or minus signs attached to it.

Analogy: The Balancing Scales

Think of a formula as a pair of perfectly balanced scales. Whatever you do to one side of the equals sign, you must do to the other side to keep it balanced.

Key Takeaway: The subject is the "lonely letter" sitting all by itself on one side of the equals sign with a \(+1\) in front of it.

2. The Golden Tool: Inverse Operations

To move things away from the letter you want to isolate, you use inverse operations (opposite operations) to "undo" what has been done:

  • Addition (\(+\)) is undone by Subtraction (\(-\))
  • Subtraction (\(-\)) is undone by Addition (\(+\))
  • Multiplication (\(\times\)) is undone by Division (\(\div\) or a fraction bar)
  • Division (\(\div\)) is undone by Multiplication (\(\times\))
  • Squaring (\(x^2\)) is undone by taking the Square Root (\(\sqrt{x}\))

Memory Trick: Working in Reverse Order

When you evaluate an expression, you follow the order of operations. When you rearrange or undo an expression, you generally work backwards: deal with any additions and subtractions first, then multiplications and divisions, and finally powers or roots!

Key Takeaway: Whatever operation is attached to your target variable, use its exact opposite on both sides to undo it.

3. Step-by-Step Rearranging: From Simple to Advanced

Type 1: One-Step and Two-Step Formulae

Example 1 (One-step): Make \(b\) the subject of \(a = b + c\).

Step 1: Look at \(b\). It has \(+ c\) attached to it.
Step 2: Subtract \(c\) from both sides: \(a - c = b\).
Step 3: Write it conventionally with the subject on the left: \(b = a - c\).

Example 2 (One-step): Make \(r\) the subject of \(C = 2\pi r\).

Step 1: Look at \(r\). It is multiplied by \(2\pi\).
Step 2: Divide both sides by \(2\pi\): \(\frac{C}{2\pi} = r\).
Step 3: Write with \(r\) on the left: \(r = \frac{C}{2\pi}\).

Example 3 (Two-step): Make \(x\) the subject of \(y = ax + b\).

Step 1: Undo the addition first by subtracting \(b\) from both sides: \(y - b = ax\).
Step 2: Undo the multiplication by dividing the entire opposite side by \(a\): \(\frac{y - b}{a} = x\).
Step 3: State your final answer: \(x = \frac{y - b}{a}\).

Type 2: Formulae Involving Fractions and Division

When a formula has a fraction, a great first move is often to multiply both sides by the denominator to clear the fraction.

Example 4: Make \(d\) the subject of \(s = \frac{d}{t}\), and then make \(t\) the subject.

To make \(d\) the subject:
Multiply both sides by \(t\): \(st = d \implies d = st\).

To make \(t\) the subject:
Start from \(st = d\). Now divide both sides by \(s\): \(t = \frac{d}{s}\).

Example 5: Make \(x\) the subject of \(y = \frac{5x + 6}{2}\).

Step 1: Clear the fraction by multiplying both sides by \(2\): \(2y = 5x + 6\).
Step 2: Subtract \(6\) from both sides: \(2y - 6 = 5x\).
Step 3: Divide everything by \(5\): \(x = \frac{2y - 6}{5}\).

Note on equivalent forms: Writing \(x = \frac{2y - 6}{5}\), \(x = \frac{2y}{5} - \frac{6}{5}\), or \(x = \frac{2(y - 3)}{5}\) are all mathematically valid and correct!

Type 3: Formulae Involving Brackets

When you see brackets, you have two choices: expand the brackets first, or divide by the multiplier outside.

Example 6: Make \(a\) the subject of \(A = h(a + b)\).

Method (Dividing first):
Step 1: Divide both sides by \(h\): \(\frac{A}{h} = a + b\).
Step 2: Subtract \(b\) from both sides: \(a = \frac{A}{h} - b\).

Type 4: Formulae Involving Powers and Roots

Don't worry if this seems tricky at first—just remember that square roots undo squares!

Example 7: Make \(r\) the subject of \(A = \pi r^2\).

Step 1: Isolate \(r^2\) first by dividing both sides by \(\pi\): \(\frac{A}{\pi} = r^2\).
Step 2: Take the square root of both sides to undo the square: \(r = \sqrt{\frac{A}{\pi}}\).

Key Takeaway: Always isolate the power term (like \(r^2\)) completely before taking the square root!

4. Important Rules & Formatting Standards

  • Left-Hand Side Preference: It is standard practice to write the new subject on the left (e.g., \(x = \dots\)), although mathematically \(\text{expression} = x\) means the exact same thing.
  • Fraction Bars Group Everything: When you divide an expression like \(y - b\) by \(a\), draw the fraction bar under the whole expression: \(x = \frac{y - b}{a}\), never \(y - \frac{b}{a}\).
  • Clean Subject: Make sure your target letter does not have a minus sign in front of it. A subject must have a coefficient of positive \(+1\).

5. Common Mistakes to Avoid

Watch out for these common traps reported by examiners:

  1. Doing an operation to only one side: Always apply changes to both sides of the equals sign to keep the balance.
  2. Partial Division: When rearranging \(y = ax + b\), writing \(x = \frac{y}{a} - b\) is incorrect. The entire side must be divided: \(x = \frac{y - b}{a}\).
  3. Stopping too early: Leaving \(-x = 5 - y\) is not finished because the subject must be positive. Multiplying/dividing by \(-1\) gives the proper subject: \(x = y - 5\).
  4. Premature Roots: In formulae like \(E = \frac{1}{2}mv^2\), never take the square root first. You must isolate \(v^2\) first (\(v^2 = \frac{2E}{m}\)) before writing \(v = \sqrt{\frac{2E}{m}}\).
  5. The Denominator Trap: If the target letter is in the denominator (like \(t\) in \(s = \frac{d}{t}\)), do not try to subtract \(d\). Multiply by \(t\) first to get it out of the denominator (\(st = d\)), then divide by \(s\) to get \(t = \frac{d}{s}\).

Quick Review Checklist

Before you move on, check if you can:

  • Identify the subject of a given formula.
  • Use inverse operations to undo addition, subtraction, multiplication, and division.
  • Clear fractions by multiplying by the denominator.
  • Isolate a squared variable before taking the square root.