FA5, HA5 - Changing the subject | Eduqas GCSE Mathematics

A formula may give the wrong quantity for the question you want to answer. Learn to rearrange it while keeping an equivalent relationship, including brackets and squares, and check a proposed rearrangement. FA5 and HA5 cover using formulae and changing their subject.

Undoing operations

If $y=ax+b$ , can you write a formula that calculates $x$ directly? Solving it once in letters gives a method that works for many numerical inputs.

Subject of a formula

The subject is the variable written alone on one side of a formula. In $y=ax+b$ , the subject is $y$ . Making $x$ the subject means writing $x=\text{an expression without }x$ .

Use the same equality-preserving operations as when solving a numerical equation. Work backwards from the last operation applied to the new subject. Treat the other letters as quantities, not as instructions to perform new operations.

If $ax$ is to be divided by $a$ , the condition $a\ne0$ matters. With $a=0$ , the original $y=ax+b$ becomes $y=b$ and gives no unique value for $x$ .

Worked example 1: undo multiplication and addition

Question. Make $x$ the subject of $y=ax+b$ , where $a\ne0$ .

Plan. Subtract $b$ from both sides, then divide the whole remaining side by $a$ .

\begin{aligned} y-b&=ax &&\text{subtract b from both sides}\\ \frac{y-b}{a}&=x &&\text{divide both sides by non-zero a} \end{aligned}

Answer. $x=\frac{y-b}{a}$ , for $a\ne0$ .

Check. With $a=3$ , $b=5$ and $y=26$ , the new formula gives $x=(26-5)/3=7$ . The original gives $3(7)+5=26$ .

Now change multiplication by $a$ into division by $a$ . The inverse operation changes with it.

Worked example 2: undo division and addition

Question. Make $x$ the subject of $y=\frac{x}{a}+b$ , where $a\ne0$ .

Plan. Subtract $b$ , then multiply both sides by $a$ .

\begin{aligned} y-b&=\frac{x}{a} &&\text{subtract b}\\ a(y-b)&=x &&\text{multiply the whole side by a} \end{aligned}

Answer. $x=a(y-b)$ , with the original condition $a\ne0$ .

Check. With $a=3$ , $b=5$ and $y=26$ , $x=3(26-5)=63$ . The original gives $63/3+5=26$ .

The brackets in the second answer keep the subtraction together. Expanding gives $ay-ab$ , so both terms are multiplied.

Sometimes the new subject starts in a denominator. For example, from $v=d/t$ , multiply by $t$ to get $vt=d$ , then divide by $v$ to get $t=d/v$ . This requires $v\ne0$ , and the original requires $t\ne0$ . The operations still act on both sides; letters do not simply “move”.

Watch out

From $y=x/a+b$ , the answer is $x=a(y-b)$ , not $ay-b$ . Multiplication by $a$ applies to the entire expression $y-b$ .

Keep the subtraction grouped in this new formula.

Brackets and powers

An inverse operation must undo the whole bracket or power it is attached to.

For example, to isolate $x$ in $Q=3(x+2)$ , divide by $3$ first to obtain $Q/3=x+2$ , then subtract $2$ . Expanding first also works, but undoing the outer multiplication keeps the steps short.

A square needs extra care: both $5^2$ and $(-5)^2$ equal $25$ . The symbol $\sqrt{25}$ means the non-negative square root, $5$ . An unrestricted equation $u^2=25$ has two solutions, $u=5$ and $u=-5$ . A context can decide which is allowed.

Square roots and conditions

For a real number $A\ge0$ , $\sqrt A$ is its non-negative square root. If $u^2=A$ , then $u=\pm\sqrt A$ ; if $u$ is known to be non-negative, only $u=\sqrt A$ is allowed.

Use the units and the quantity's meaning to choose a root. A side length cannot be negative.

Worked example 3: a squared length

Question. A square has area $A\,\mathrm{cm}^2$ and side length $x\,\mathrm{cm}$ , where $x>0$ . Make $x$ the subject of $A=x^2$ .

Plan. Take a square root and use the positive-length condition.

\begin{aligned} x^2&=A &&\text{read the equality the other way}\\ x&=\sqrt A &&\text{choose the positive length} \end{aligned}

Answer. $x=\sqrt A$ , with $A>0$ .

Check. For $A=49$ , $x=7$ , and $7^2=49$ . The algebraic value $-7$ also squares to $49$ , but cannot be a side length.

Now the whole side is $x+2$ . Take the square root before subtracting $2$ .

Continue this lesson.

You’re previewing NovaLearn without an account. Create a free account to read the rest.

Create free account

Last Accessed

Number

Algebra

Ratio and proportion

Geometry and measures

Higher graphs

Probability

Statistics

FA5, HA5 - Changing the subject

Undoing operations

Brackets and powers

Continue this lesson.

Learn faster with Nova