Prepared by Muhammad Tayyab, Subject Specialist Mathematics, Govt Christian High School Daska
📌 Based on National Curriculum 2023 / PECTAA 2026 Syllabus
📖 What's Inside: This exercise covers making a variable the subject of a formula – rearranging linear, quadratic, and geometric formulas to solve for a specific variable. Perfect for Punjab Boards exam preparation.
📚 Related Resources – Unit 2: Quadratic Equations & Inequalities
1
Make \(F\) the subject of the formula, \(\displaystyle C^\circ = \frac{5}{9}(F^\circ - 32)\).
\[\begin{aligned} C^\circ &= \frac{5}{9}(F^\circ - 32) \\[6pt] C^\circ \cdot \frac{9}{5} &= F^\circ - 32 \\[6pt] \frac{9C^\circ}{5} + 32 &= F^\circ \\[6pt] \boldsymbol{F^\circ} &= \boldsymbol{\dfrac{9C^\circ}{5} + 32} \end{aligned}\]
2
The formula for finding simple interest is \(I = PRT\).
(a) Make \(P\) the subject of the formula.
\[\begin{aligned} I &= PRT \\ \frac{I}{RT} &= P \\[4pt] \boldsymbol{P} &= \boldsymbol{\dfrac{I}{RT}} \end{aligned}\]
(b) Make \(T\) the subject of the formula.
\[\begin{aligned} I &= PRT \\ \frac{I}{PR} &= T \\[4pt] \boldsymbol{T} &= \boldsymbol{\dfrac{I}{PR}} \end{aligned}\]
3
Make \(a\) the subject of the formula \(S = 2a + (n-1)d\).
\[\begin{aligned} S &= 2a + (n-1)d \\ S - (n-1)d &= 2a \\ \frac{S-(n-1)d}{2} &= a \\[4pt] \boldsymbol{a} &= \boldsymbol{\dfrac{S-(n-1)d}{2}} \end{aligned}\]
4
The volume of a cylinder is given by the formula \(V = \pi r^2 h\). Make \(h\) the subject of the formula.
\[\begin{aligned} V &= \pi r^2 h \\ \frac{V}{\pi r^2} &= h \\[4pt] \boldsymbol{h} &= \boldsymbol{\dfrac{V}{\pi r^2}} \end{aligned}\]
5
The area of a trapezoid is \(\displaystyle A = \frac{1}{2}h(b_1 + b_2)\), make \(h\) the subject of the formula.
\[\begin{aligned} A &= \frac{1}{2}h(b_1+b_2) \\ 2A &= h(b_1+b_2) \\ \frac{2A}{b_1+b_2} &= h \\[4pt] \boldsymbol{h} &= \boldsymbol{\dfrac{2A}{b_1+b_2}} \end{aligned}\]
6
If \(y = mx + c\), then make \(x\) the subject of this equation.
\[\begin{aligned} y &= mx + c \\ y - c &= mx \\ \frac{y-c}{m} &= x \\[4pt] \boldsymbol{x} &= \boldsymbol{\dfrac{y-c}{m}} \end{aligned}\]
7
Perimeter \((P)\) of a rectangle is \(P = 2(l + w)\), make \(l\) as the subject of this formula.
\[\begin{aligned} P &= 2(l+w) \\ P &= 2l+2w \\ P+2w &= 2l \\ \frac{P+2w}{2} &= l \\[4pt] \boldsymbol{l} &= \boldsymbol{\dfrac{P+2w}{2}} \end{aligned}\]
8
The equation of a parabola is \(y^2 = 4ax\), make \(x\) as a subject of this equation.
\[\begin{aligned} y^2 &= 4ax \\ \frac{y^2}{4a} &= x \\[4pt] \boldsymbol{x} &= \boldsymbol{\dfrac{y^2}{4a}} \end{aligned}\]
9
If \(P = S - C\), where \(S\) is selling price and \(C\) is cost price. Make \(S\) as subject of the equation.
\[\begin{aligned} P &= S - C \\ P+C &= S \\[4pt] \boldsymbol{S} &= \boldsymbol{P+C} \end{aligned}\]
10
Volume of the cone is \(\displaystyle V = \frac{1}{3}\pi r^2 h\), make \(h\) as subject of this formula.
\[\begin{aligned} V &= \frac{1}{3}\pi r^2 h \\ 3V &= \pi r^2 h \\ \frac{3V}{\pi r^2} &= h \\[4pt] \boldsymbol{h} &= \boldsymbol{\dfrac{3V}{\pi r^2}} \end{aligned}\]
📈 Key Concepts – Changing the Subject
- Goal: Isolate the required variable on one side of the equation.
- Inverse Operations: Use addition/subtraction, multiplication/division, and powers/roots to rearrange.
- Order of Operations: Work from the outside in – undo addition/subtraction first, then multiplication/division.
- Check: Verify by substituting a simple value into the original and rearranged formulas.