S4 Mathematics - Term 1

Topic 2: Equations and Inequalities

Lesson 4: Complex Subject Changes

Duration: 40 minutes

Learning Outcomes

By the end of this lesson, you should be able to:

• Rearrange formulas requiring two or more steps
• Change the subject when the variable appears in a fraction
• Handle formulas involving squares and square roots

Quick Review

From Lesson 3:

Make x the subject of y = 3x

Answer: x = y/3 (divide both sides by 3)

What if there are two operations to undo?

Two Operations

Formula: y = 2x + 5

What operations are done to x?

1. First: multiply by 2
2. Then: add 5

To undo, we work backwards!

Working Backwards

Example 1: y = 2x + 5

Make x the subject

Step 1: Undo "add 5" - subtract 5 from both sides
y - 5 = 2x + 5 - 5
y - 5 = 2x

Step 2: Undo "multiply by 2" - divide both sides by 2
(y - 5)/2 = x

x = (y - 5)/2

Example 2: v = u + at

Make a the subject

Operations on a: multiply by t, then add u

Step 1: Subtract u from both sides
v - u = at

Step 2: Divide both sides by t
(v - u)/t = a

a = (v - u)/t

Example 3: P = 2l + 2w

Make l the subject

Operations on l: multiply by 2, then add 2w

Step 1: Subtract 2w
P - 2w = 2l

Step 2: Divide by 2
(P - 2w)/2 = l

l = (P - 2w)/2

Pair Practice

Make the indicated variable the subject:

1. a = 3b - 7 (make b the subject)
2. y = (x + 4)/2 (make x the subject)

Answers

1. a = 3b - 7

   • Add 7: a + 7 = 3b
   • Divide by 3: b = (a + 7)/3

2. y = (x + 4)/2

   • Multiply by 2: 2y = x + 4
   • Subtract 4: x = 2y - 4

Variable in Denominator

Formula: s = d/t

Make t the subject

t is in the denominator, so first multiply both sides by t

st = d

Then divide by s: t = d/s

Formulas with Squares

Formula: A = pi r squared

Make r the subject

Operations on r: square, then multiply by pi

Rearranging with Squares

Example: A = pi r squared

Step 1: Divide by pi
A/pi = r squared

Step 2: Take the square root
sqrt(A/pi) = r

r = sqrt(A/pi)

Remember: Square root undoes squaring!

Example: E = (1/2)mv squared

Make v the subject

Step 1: Multiply by 2
2E = mv squared

Step 2: Divide by m
2E/m = v squared

Step 3: Take the square root
v = sqrt(2E/m)

Individual Practice

Make the indicated variable the subject:

1. C = 2 pi r (make r)
2. V = pi r squared h (make h)
3. v squared = u squared + 2as (make u)

Answers

1. C = 2 pi r
   r = C/(2 pi)

2. V = pi r squared h
   h = V/(pi r squared)

3. v squared = u squared + 2as

   • Subtract 2as: v squared - 2as = u squared
   • Square root: u = sqrt(v squared - 2as)

Summary

Multi-step rearrangement process:

1. Identify all operations on the variable
2. List them in order (using BODMAS)
3. Undo them in reverse order
4. Apply inverse operations to both sides

Key Strategies

Exit Questions

Make the indicated variable the subject:

1. y = 4x - 3 (make x)
2. A = pi r squared (make r)
3. s = ut + (1/2)at squared (make u)

Answers

1. y = 4x - 3
   x = (y + 3)/4

2. A = pi r squared
   r = sqrt(A/pi)

3. s = ut + (1/2)at squared

   • Subtract (1/2)at squared: s - (1/2)at squared = ut
   • Divide by t: u = (s - (1/2)at squared)/t

Homework

Make the indicated variable the subject:

1. y = 5x + 2 (make x)
2. F = ma (make m)
3. v = u + at (make t)
4. A = (1/2)bh (make h)
5. E = (1/2)mv squared (make m)
6. v squared = u squared + 2as (make s)

Next Lesson

Lesson 5: Solving Linear Equations Review

We will apply our skills to:

• Solve equations step by step
• Check solutions by substitution
• Work with word problems

Credits

Created: December 2025
Based on: NCDC Lower Secondary Mathematics Syllabus (2019)

Source: National Curriculum Development Centre (NCDC), Uganda