Changing the subject where the variable appears in multiple terms | Maths

Starter quiz

In which of these is $x$ the subject?

$x = \frac{xy+4}{2}$

${1\over x} = {1\over y+2}$

$\frac{2y+2z}{3y+3z}=x$ ✓

$x=2(3z+y) + \frac{z}{y}$ ✓

$\frac{3w(2+y)}{x(2 + y)}=x$
Make $x$ the subject of $3x+4=5x-b$ .

$x = 2 + b$

$x=\frac{4+b}{2}$ ✓

$x=\frac{4-b}{2}$

$x=\frac{3x+4+b}{5}$

$x=\frac{5x-4-b}{3}$
Factorise $x^2 - 6x + 8$ .

$(x-8)(x-1)$

$(x-8)(x+1)$

$(x-6)(x+8)$

$(x-4)(x-2)$ ✓

$(x-4)(x+2)$
Shown is the formula for the perimeter of a rectangle. Make $b$ the subject.

$b = P - a$

$b=2P-a$

$2b=P-2a$

$b=\frac{P+2a}{2}$

$b=\frac{P}{2}-a$ ✓
Make $z$ the subject of $\frac{4}{x} +\frac{2}{y} = \frac{3}{z}$ .

$z=\frac{x+y}{2}$

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

$z=\frac{3x+6y}{4}$

$z=\frac{3xy}{4y+2x}$ ✓

$z= 3-\frac{4y+2x}{xy}$
Simplify $\frac{3x-12}{2x-6} + \frac{x}{3x-9}$

$\frac{7x^2-7x+9}{6x-18}$

$\frac{11x-36}{6x-18}$ ✓

$\frac{11x-4}{6x-3}$

$\frac{2x+9}{6}$

$\frac{4}{5}$

Exit quiz

What is the most efficient first step to make $x$ the subject of $ax + b = 2x + c$ ?

Divide both sides by 2.

Factorise both sides of the equation.

Subtract $2x$ from both sides. ✓
What is the most efficient next step to make $x$ the subject of $ax-2x = c-b$ ?

Add $b$ to both sides.

Divide both sides by 2.

Divide both sides by $a$ .

Factorise the expressions. ✓
Make $a$ the subject of $a(b+c)=3+4(a+2b)$ .

$a=\frac{3+8b}{b+c-4}$ ✓

$a=\frac{3+8b}{b+c+4}$

$a=\frac{14b}{b+c-7}$

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

$a=\frac{3+4a+8b}{b+c}$
Make $r$ the subject of $p = \frac{r+5}{r+q}$ .

$r=\frac{pq}{5}$

$r=\frac{r+5}{p}-q$

$r=\frac{5+pq}{p}$

$r=\frac{5-q}{p-1}$

$r=\frac{5-pq}{p-1}$ ✓
What would be the most efficient first step to make $a$ the subject of $\frac{8a+4b}{6a+3b}+\frac{a+1}{3a+3b}=b$ ?

Factorise and look for common factors. ✓

Add the 2 fractions.

Multiply both sides by $6a+3b$ .

Subtract $b$ from both sides.
Make $a$ the subject of $\frac{4(2a+b)}{3(2a+b)}+\frac{a+1}{3(a+b)}=b$ .

$a=\frac{3b^2-2b-1}{5-3b}$

$a=\frac{3b^2-4b-1}{5-3b}$ ✓

$a=\frac{3b^2-2b-1}{9-6b}$

$a=\frac{9b^2-4b-1}{9-7b}$

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Lesson Details

Key learning points

The variable to become the subject may appear in more than one term.
It is possible the terms cannot be additively combined.
In this situation, it is possible to factorise with one of the factors being the desired variable.

Common misconception

Not fully isolating the required variable by leaving other terms containing the variable in the expression.

To make a variable the subject, it must be expressed in terms of other variables only. Factorising helps isolate a variable if it is in multiple terms.

Keywords

Subject of an equation/formula - The subject of an equation/a formula is a variable that is expressed in terms of other variables. It should have an exponent of 1 and a coefficient of 1