Change of Subject Formulae

Lesson Objectives

Define and explain the concept of changing the subject of a formula.
Manipulate algebraic formulas to make a specified variable the subject.
Apply inverse operations in formula rearrangement.
Solve real-world problems using rearranged formulas.
Identify and correct errors in incorrect rearrangements.

Lesson Introduction

In real-world scenarios like engineering and physics, we often work with formulas where we need to isolate one variable. For example, in motion equations, we might want to find time instead of distance. Changing the subject of a formula equips you with the algebraic tools to rearrange equations to solve for different variables.

Core Lesson Content

To change the subject of a formula means to rearrange the equation to make a different variable the subject. This is often done by performing inverse operations to isolate the desired variable.

Worked Examples

Example 1 (Basic):
Make

x

the subject of

y = x + 5

Subtract 5 from both sides:

x = y - 5

Example 2 (Intermediate):
Make

x

the subject of

y = 3x - 2

Add 2:

y + 2 = 3x

Divide by 3:

x = \frac{y + 2}{3}

Example 3 (With fractions):
Make

x

the subject of

y = \frac{2x + 1}{5}

Multiply both sides by 5:

5y = 2x + 1

Subtract 1:

5y - 1 = 2x

Divide by 2:

x = \frac{5y - 1}{2}

Example 4 (Advanced):
Make

x

the subject of

A = \frac{x - 3}{x + 2}

Multiply both sides by

(x + 2)

A(x + 2) = x - 3

Expand:

Ax + 2A = x - 3

Rearranged:

Ax - x = -3 - 2A

Factor:

x(A - 1) = -3 - 2A

Final answer:

x = \frac{-3 - 2A}{A - 1}

Example 5 (Formula):
Make

r

the subject of

A = \pi r^2

Divide both sides by

\pi

\frac{A}{\pi} = r^2

Take square root:

r = \sqrt{\frac{A}{\pi}}

Exercises

Make $x$ the subject of $y = 2x + 7$
Make $v$ the subject of $s = vt$
Make $x$ the subject of $y = \frac{x + 4}{3}$
Make $t$ the subject of $s = ut + \frac{1}{2}at^2$
[WAEC] Make $x$ the subject of $y = \frac{2x - 1}{x + 3}$ (Past Question)
[NECO] Make $r$ the subject of $V = \frac{4}{3}\pi r^3$ (Past Question)
[JAMB] Make $x$ the subject of $A = \frac{x + a}{x - a}$ (Past Question)
Make $x$ the subject of $P = \frac{2x}{x - 1}$
Make $m$ the subject of $y = mx + c$
[NABTEC] Make $x$ the subject of $y = \sqrt{3x + 1}$ (Past Question)

Conclusion / Recap

In this lesson, you’ve learned how to rearrange formulas to make different variables the subject using algebraic operations. This is a critical skill in many subjects, especially in sciences and applied mathematics.
Next up: Simultaneous Equations – Solving Two Equations Together!