Algebra · Junior Cert

Change of Subject

Junior Cert Higher · Rearranging formulae & square roots · Tap NEXT to begin

Section 1 of 3

The Idea

Changing the subject means rearranging a formula so a different letter is on its own. You undo the operations, exactly like solving — but the answer is a formula, not a number.

Worked example — a number first

Find $x$ when $2x+3 = 11$.

$2x = 8$

$x = 4$

Worked example — now with letters

Find $x$ when $ax+b = c$.

$ax = c-b$

$x = \dfrac{c-b}{a}$

Worked example

Find $x$ when $xy = A$.

$x = \dfrac{A}{y}$

Worked example — two ways

Find $x$ when $a(x+y) = b$.

Multiply out: $ax+ay = b \Rightarrow ax = b-ay$

$x = \dfrac{b-ay}{a}$

Section 2 of 3

Make $x$ the Subject

Worked example

Make $x$ the subject of $px-q = r$.

$px = r+q$

$x = \dfrac{r+q}{p}$

Worked example

Make $x$ the subject of $a(x+c) = q$.

$ax+ac = q \Rightarrow ax = q-ac$

$x = \dfrac{q-ac}{a}$

Worked example

Make $x$ the subject of $12x+3 = 5x+11$.

$12x-5x = 11-3 \Rightarrow 7x = 8$

$x = \dfrac{8}{7}$

Worked example — $x$ on both sides

Make $x$ the subject of $px+q = rx+s$.

$px-rx = s-q$

$(p-r)x = s-q$

$x = \dfrac{s-q}{p-r}$

Worked example

Make $x$ the subject of $b(x+3) = a(x+5)$.

$bx+3b = ax+5a$

$bx-ax = 5a-3b \Rightarrow (b-a)x = 5a-3b$

$x = \dfrac{5a-3b}{b-a}$

Worked example — a fraction

Make $x$ the subject of $\dfrac{a}{x+3} = \dfrac{b}{x-1}$.

Cross-multiply: $a(x-1) = b(x+3)$

$ax-a = bx+3b \Rightarrow ax-bx = 3b+a$

$(a-b)x = 3b+a$

$x = \dfrac{3b+a}{a-b}$

You try

Make $x$ the subject of $\dfrac{x+3}{a} = \dfrac{x+11}{b}$.

Pen and paper out — try it before you reveal.

Cross-multiply: $b(x+3) = a(x+11)$

$bx+3b = ax+11a \Rightarrow (b-a)x = 11a-3b$

$x = \dfrac{11a-3b}{b-a}$

You try

Make $x$ the subject of $\dfrac{p}{x-5} = \dfrac{q}{x-7}$.

Pen and paper out — try it before you reveal.

$p(x-7) = q(x-5)$

$px-7p = qx-5q \Rightarrow (p-q)x = 7p-5q$

$x = \dfrac{7p-5q}{p-q}$

You try

Make $x$ the subject of $\dfrac{a}{2x-1} = \dfrac{b}{3x-7}$.

Pen and paper out — try it before you reveal.

$a(3x-7) = b(2x-1)$

$3ax-7a = 2bx-b \Rightarrow (3a-2b)x = 7a-b$

$x = \dfrac{7a-b}{3a-2b}$

Section 3 of 3

Other Letters & Square Roots

Worked example — make $p$ the subject

Make $p$ the subject of $a(p+b) = c(p+d)$.

$ap+ab = cp+cd$

$ap-cp = cd-ab \Rightarrow p(a-c) = cd-ab$

$p = \dfrac{cd-ab}{a-c}$

Worked example — express $a$ in terms of $b$ and $c$

Given $\dfrac{1}{b} = \dfrac{1}{a}+\dfrac{1}{c}$, write $a$ in terms of $b$ and $c$.

Over a common bottom: $ac = bc+ab$

$ac-ab = bc \Rightarrow a(c-b) = bc$

$a = \dfrac{bc}{c-b}$

Worked example

Given $\dfrac{2a+3b}{4} = c$, write $a$ in terms of $b$ and $c$.

$2a+3b = 4c \Rightarrow 2a = 4c-3b$

$a = \dfrac{4c-3b}{2}$

You try

Given $\dfrac{6a-3b}{5} = \dfrac{c}{2}$, write $a$ in terms of $b$ and $c$.

Pen and paper out — try it before you reveal.

Cross-multiply: $2(6a-3b) = 5c$

$12a-6b = 5c \Rightarrow 12a = 5c+6b$

$a = \dfrac{5c+6b}{12}$

Square roots. To undo a square root, square both sides. Remember $\sqrt{25}=5$ and $\sqrt{100}=10$.

Worked example

Given $\sqrt{a+3} = b$, write $a$ in terms of $b$.

Square both sides: $a+3 = b^2$

$a = b^2-3$

Worked example

Given $\sqrt{ax+b} = c$, write $x$ in terms of $a,b,c$.

$ax+b = c^2 \Rightarrow ax = c^2-b$

$x = \dfrac{c^2-b}{a}$

Worked example

Solve $\sqrt{x}+3 = 8$.

$\sqrt{x} = 5$

Square: $x = 25$

$x = 25$

You try

Solve $\sqrt{x}+2 = 9$.

Pen and paper out — try it before you reveal.

$\sqrt{x} = 7$

Square both sides

$x = 49$

Worked example

Given $\sqrt{a^2x-b} = c$, write $a$ in terms of $b,c,x$.

$a^2x-b = c^2 \Rightarrow a^2x = c^2+b$

$a^2 = \dfrac{c^2+b}{x}$

$a = \sqrt{\dfrac{c^2+b}{x}}$

You try

Given $\sqrt{a^2x}+b = c$, write $a$ in terms of $b,c,x$.

Pen and paper out — try it before you reveal.

$\sqrt{a^2x} = c-b$

Square: $a^2x = (c-b)^2 \Rightarrow a^2 = \dfrac{(c-b)^2}{x}$

$a = \sqrt{\dfrac{(c-b)^2}{x}}$

Careful: $\sqrt{a+b} \ne \sqrt{a}+\sqrt{b}$. For example $\sqrt{4}+\sqrt{9} = 2+3 = 5$, but $\sqrt{4+9} = \sqrt{13} \approx 3.6$.

That’s change of subject.

Undo the operations to free the letter you want — multiply out, gather it, factor it out, divide, and square to kill a root.