A formula is an equation that shows the relationship between different variables (e.g., Area = length × width). This lesson focuses on two essential skills: changing the subject of a formula and substitution, with a special focus on formulas involving squares and square roots.

1. Core Concepts

• Subject of a Formula: The single variable isolated on one side of the equals sign. In A = πr², the subject is A.

• Changing the Subject: Rearranging the formula to isolate a different variable. This is a crucial algebra skill.

• Inverse Operations: The key to rearranging. To move something to the other side of the equation, you do the opposite operation.

  • Addition ↔ Subtraction

  • Multiplication ↔ Division

  • Squaring (x²) ↔ Taking the Square Root (√x)

2. Key Methods

Part A: Changing the Subject

The goal is to "unwrap" the variable you want to make the subject. Work from the outside in, reversing each operation.

Scenario 1: The new subject is SQUARED

• Example: Make r the subject of A = πr².

• Steps:

  1. Isolate the squared term (r²): The r² is multiplied by π. The inverse is division.

     • A / π = r²

  2. Undo the square: The inverse of squaring is taking the square root.

     • √(A / π) = r

  3. Final Answer: r = √(A / π)

Scenario 2: The new subject is under a SQUARE ROOT

• Example: Make l the subject of t = 2π√(l/g).

• Steps:

  1. Isolate the square root term: The √ term is multiplied by 2π. The inverse is division.

     • t / (2π) = √(l/g)

  2. Undo the square root: The inverse of a square root is squaring. You must square the entire other side.

     • (t / (2π))² = l/g

     • t² / (4π²) = l/g

  3. Isolate l: The l is divided by g. The inverse is multiplication.

     • (g * t²) / (4π²) = l

  4. Final Answer: l = (gt²) / (4π²)

Part B: Substitution to Find a Value

You have two valid methods. Choose the one you find easier.

• Method 1: Substitute First, then Solve (Often easier)

• Method 2: Rearrange First, then Substitute

Example: Using v² = u² + 2as, find the value of u when v=10, a=3, and s=6.

• Using Method 1 (Substitute First):

  1. Substitute the numbers:

     • 10² = u² + 2(3)(6)

  2. Simplify:

     • 100 = u² + 36

  3. Solve the simple equation for u:

     • 100 - 36 = u²

     • 64 = u²

     • u = √64

     • u = 8

3. Exam Tips & Tricks

• The ± Sign: Mathematically, √64 is +8 or -8. However, in formula questions, variables often represent physical quantities like length, radius, or time, which cannot be negative. In these cases, you only take the positive root. Read the context!

• Unwrap the Subject Carefully: When rearranging, deal with the terms "furthest" from your target variable first.

  • For v² = u² + 2as, to make a the subject:

    1. First, move the u² term (subtraction).

    2. Then, move the 2s term (division).

• Squaring Both Sides: When you square a side, ensure you square the entire side, not just individual parts. If you have t / (2π), the square is t² / (4π²), not t² / (2π).

4. Important Points to Remember

• Show your rearrangement steps clearly. You can get method marks even if your final calculation is wrong.

• Substitution problems are often linked to other topics (e.g., finding the radius of a cylinder when the volume is known).

• Be methodical and perform one inverse operation at a time to avoid errors.