1.2.3 Change the subject of a formula

What you'll learn

What the subject of a formula is.
How inverse operations let you isolate a chosen variable.
How to rearrange formulas involving fractions, roots and powers.
What to do when the required subject appears more than once.

Formulae and subjects

A formula is an equation showing a relationship between two or more variables. For example,

v = u + a t

relates the variables $v$ , $u$ , $a$ and $t$ .

Definition

Subject of a formula

The subject is the variable written alone on one side of the equals sign. In $v = u + a t$ , the subject is $v$ .

To change the subject means to rearrange the formula so that a different variable is alone. For example, changing the subject of $v = u + a t$ to $t$ gives

t=v−ua.t=\frac{v-u}{a}.

The formula still describes exactly the same relationship. Only its arrangement has changed.

Maintaining equality

An equation is a statement that its two sides have equal values. When rearranging it, you must perform the same valid operation on both sides.

Key Idea

Keep the equation balanced

Whatever you do to one side of an equation, do the same to the other side. This preserves equality.

In written working, you will often simplify both sides immediately rather than explicitly showing the operation twice. For instance,

y + 5 = x

becomes $y = x - 5$ after subtracting 5 from both sides.

Inverse operations

Inverse operations undo one another:

addition and subtraction are inverses;
multiplication and division are inverses;
squaring and taking a square root are inverses;
raising to the power $n$ and taking the $n$ th root are inverses.

You usually undo operations in the reverse order from that in which they act on the required variable.

Example

Making a variable the subject

Make $x$ the subject of

y = 5 x - 7.

The subtraction of 7 is the outermost operation acting on $x$ , so add 7 to both sides:
$y + 7 = 5 x .$
Divide both sides by 5 to undo the multiplication:
$x=y+75.x=\frac{y+7}{5}.$
Check by substituting the result into the original right-hand side:
$5(y+75)−7=y+7−7=y.5\left(\frac{y+7}{5}\right)-7=y+7-7=y.$
This recovers the original left-hand side, so the rearrangement is consistent.

Common Mistake

Moving terms without changing signs

Do not think of a term as simply “moving across” the equals sign. Its sign changes because you have applied an inverse operation to both sides.

Rearranging formulas with brackets

If the required subject is inside a bracket, you can sometimes divide by the factor outside the bracket before expanding. This is often shorter and reduces the chance of an algebraic error.

Example

Rearranging without expanding

Make $r$ the subject of

P = 2 (l + r) .

Divide both sides by 2 to isolate the bracket:
$P2=l+r.\frac{P}{2}=l+r.$
Subtract $l$ from both sides:
$r=P2−l.r=\frac{P}{2}-l.$

Tip

Keep useful brackets intact

Before expanding a bracket, ask whether dividing by its outside factor will isolate it. Preserving the bracket often makes the structure of the formula easier to see.

Formulas involving fractions

When the required subject appears in a denominator, first remove the fraction by multiplying both sides by the denominator. This is sometimes called clearing the denominator.

Example

Removing a denominator

Make $t$ the subject of

q=pt+3.q=\frac{p}{t+3}.

Multiply both sides by $t + 3$ :
$q (t + 3) = p .$
Divide by $q$ :
$t+3=pq.t+3=\frac{p}{q}.$
Subtract 3:
$t=pq−3.t=\frac{p}{q}-3.$

This rearrangement assumes that the divisions used are valid. In particular, $q≠0q\neq 0$ .

Common Mistake

Restrictions from denominators

A denominator cannot equal zero. In the original formula above, $t≠−3t\neq -3$ . Rearranging a formula does not remove restrictions that were present in the original.

A subject in the numerator and denominator

You may need to multiply out brackets and then collect all terms containing the required subject.

Example

Rearranging a rational formula

Make $x$ the subject of

y=3x+1x−2.y=\frac{3x+1}{x-2}.

Multiply both sides by $x - 2$ :
$y (x - 2) = 3 x + 1.$
Expand and collect the terms containing $x$ on one side:
$yx−2y=3x+1yx−3x=2y+1.\begin{aligned} yx-2y&=3x+1\\ yx-3x&=2y+1. \end{aligned}$
Factorise the left-hand side by taking out the common factor $x$ :
$x (y - 3) = 2 y + 1.$
Divide by $y - 3$ :
$x=2y+1y−3.x=\frac{2y+1}{y-3}.$

Here $x≠2x\neq 2$ in the original formula, and the final expression also requires $y≠3y\neq 3$ . In fact, the original formula can never produce $y = 3$ .

When the subject appears more than once

If the required variable appears in several terms, you cannot usually isolate each occurrence separately. Instead:

collect every term containing the variable on one side;
move the remaining terms to the other side;
factorise by taking out the required variable;
divide by the remaining factor.

Key Idea

Collect, factorise, divide

When the new subject appears in more than one term, factorisation turns its several appearances into one common factor that can be isolated.

Example

Collecting repeated occurrences

Make $x$ the subject of

a (x + 2) = b (x - 1) .

Expand both sides so that the terms containing $x$ can be collected:
$a x + 2 a = b x - b .$
Put the $x$ -terms on the same side and the other terms on the opposite side:
$a x - b x = - b - 2 a .$
Factorise the left-hand side:
$x (a - b) = - (b + 2 a) .$
Divide by $a - b$ and simplify the signs:
$x=b+2ab−a.x=\frac{b+2a}{b-a}.$

Common Mistake

Dividing before factorising

From $a x - b x = - b - 2 a$ , do not try to divide the two $x$ -terms separately. First write $x (a - b) = - b - 2 a$ , then divide by the complete coefficient $a - b$ .

Powers and roots

When a power acts on an expression containing the required subject, isolate that powered expression before applying the inverse power.

Example

Undoing a square

Make $r$ the subject of

A=πr2.A=\pi r^2.

Divide by $π\pi$ to isolate $r^2$ :
$r2=Aπ.r^2=\frac{A}{\pi}.$
Taking square roots gives
$r=±Aπ.r=\pm\sqrt{\frac{A}{\pi}}.$
If $r$ represents a radius, the model requires $r≥0r\ge 0$ , so use
$r=Aπ.r=\sqrt{\frac{A}{\pi}}.$

The algebraic equation has both positive and negative solutions, but the meaning of a variable may restrict which values are sensible.

Common Mistake

Forgetting the negative root

Algebraically, if $x^2=k$ with $k > 0$ , then $x=±kx=\pm\sqrt{k}$ . Keep both signs unless the context or a stated restriction rules one out.

Fractional powers

The same method works with other powers. For example, if

V=43πr3,V=\frac{4}{3}\pi r^3,

then isolating $r^3$ and taking a cube root gives

r=3V4π3.r=\sqrt[3]{\frac{3V}{4\pi}}.

A cube root has one real value for every real input, so a $±\pm$ sign is not introduced.

Choosing an efficient order

More than one rearrangement route may be valid. A reliable approach is to look at the operations surrounding the required subject and undo the outermost operation first.

For example, in

y=ax+bc,y=\frac{ax+b}{c},

the expression $a x + b$ is first formed and then divided by $c$ . To recover $x$ , reverse that order: multiply by $c$ , subtract $b$ , then divide by $a$ .

Example

Reversing the order of operations

Make $x$ the subject of

y=4x−53.y=\frac{4x-5}{3}.

Multiply by 3 to undo the division:
$3 y = 4 x - 5.$
Add 5 to undo the subtraction:
$3 y + 5 = 4 x .$
Divide by 4:
$x=3y+54.x=\frac{3y+5}{4}.$

Checking a rearrangement

You can check your result in two useful ways.

Algebraic substitution: substitute your new expression back into the original formula and simplify.
Numerical check: choose permissible values for the variables, evaluate the original formula, and confirm that the rearranged formula returns the starting value.

A numerical check can expose a sign or bracket error, although it is not a proof. Algebraic substitution gives a stronger check.

Tip

Use brackets during substitution

When substituting an expression in place of a variable, put the whole expression in brackets. This prevents powers and multiplication from being applied to only part of it.

Exam technique

In the exam

Identify the variable that must become the subject and note every place where it occurs.
Preserve equality by applying inverse operations to both sides, keeping useful brackets intact where possible.
If the variable appears more than once, collect its terms, factorise and then divide.
State any important restrictions, and use the context to decide whether all algebraic solutions are valid.
Check signs, brackets and denominators by substituting your result into the original formula.

Self review

Check yourself

Can you make $h$ the subject of $V=πr2hV=\pi r^2h$ ?
Can you make $x$ the subject of $p=x+42x−1p=\frac{x+4}{2x-1}$ ?
Why does making $x$ the subject of $y=x^2+3$ usually produce two possible expressions?

What you'll learn

What the subject of a formula is.
How inverse operations let you isolate a chosen variable.
How to rearrange formulas involving fractions, roots and powers.
What to do when the required subject appears more than once.

Formulae and subjects

A formula is an equation showing a relationship between two or more variables. For example,

v = u + a t

relates the variables $v$ , $u$ , $a$ and $t$ .

Definition

Subject of a formula

The subject is the variable written alone on one side of the equals sign. In $v = u + a t$ , the subject is $v$ .

To change the subject means to rearrange the formula so that a different variable is alone. For example, changing the subject of $v = u + a t$ to $t$ gives

t=v−ua.t=\frac{v-u}{a}.

The formula still describes exactly the same relationship. Only its arrangement has changed.

Maintaining equality

An equation is a statement that its two sides have equal values. When rearranging it, you must perform the same valid operation on both sides.

Key Idea

Keep the equation balanced

Whatever you do to one side of an equation, do the same to the other side. This preserves equality.

In written working, you will often simplify both sides immediately rather than explicitly showing the operation twice. For instance,

y + 5 = x

becomes $y = x - 5$ after subtracting 5 from both sides.

Inverse operations

Inverse operations undo one another:

addition and subtraction are inverses;
multiplication and division are inverses;
squaring and taking a square root are inverses;
raising to the power $n$ and taking the $n$ th root are inverses.

You usually undo operations in the reverse order from that in which they act on the required variable.

Example

Making a variable the subject

Make $x$ the subject of

y = 5 x - 7.

The subtraction of 7 is the outermost operation acting on $x$ , so add 7 to both sides:
$y + 7 = 5 x .$
Divide both sides by 5 to undo the multiplication:
$x=y+75.x=\frac{y+7}{5}.$
Check by substituting the result into the original right-hand side:
$5(y+75)−7=y+7−7=y.5\left(\frac{y+7}{5}\right)-7=y+7-7=y.$
This recovers the original left-hand side, so the rearrangement is consistent.

Common Mistake

Moving terms without changing signs

Do not think of a term as simply “moving across” the equals sign. Its sign changes because you have applied an inverse operation to both sides.

Rearranging formulas with brackets

If the required subject is inside a bracket, you can sometimes divide by the factor outside the bracket before expanding. This is often shorter and reduces the chance of an algebraic error.

Example

Rearranging without expanding

Make $r$ the subject of

P = 2 (l + r) .

Divide both sides by 2 to isolate the bracket:
$P2=l+r.\frac{P}{2}=l+r.$
Subtract $l$ from both sides:
$r=P2−l.r=\frac{P}{2}-l.$

Tip

Keep useful brackets intact

Before expanding a bracket, ask whether dividing by its outside factor will isolate it. Preserving the bracket often makes the structure of the formula easier to see.

Formulas involving fractions

When the required subject appears in a denominator, first remove the fraction by multiplying both sides by the denominator. This is sometimes called clearing the denominator.

Example

Removing a denominator

Make $t$ the subject of

q=pt+3.q=\frac{p}{t+3}.

Multiply both sides by $t + 3$ :
$q (t + 3) = p .$
Divide by $q$ :
$t+3=pq.t+3=\frac{p}{q}.$
Subtract 3:
$t=pq−3.t=\frac{p}{q}-3.$

This rearrangement assumes that the divisions used are valid. In particular, $q≠0q\neq 0$ .

Common Mistake

Restrictions from denominators

A denominator cannot equal zero. In the original formula above, $t≠−3t\neq -3$ . Rearranging a formula does not remove restrictions that were present in the original.

A subject in the numerator and denominator

You may need to multiply out brackets and then collect all terms containing the required subject.

Example

Rearranging a rational formula

Make $x$ the subject of

y=3x+1x−2.y=\frac{3x+1}{x-2}.

Multiply both sides by $x - 2$ :
$y (x - 2) = 3 x + 1.$
Expand and collect the terms containing $x$ on one side:
$yx−2y=3x+1yx−3x=2y+1.\begin{aligned} yx-2y&=3x+1\\ yx-3x&=2y+1. \end{aligned}$
Factorise the left-hand side by taking out the common factor $x$ :
$x (y - 3) = 2 y + 1.$
Divide by $y - 3$ :
$x=2y+1y−3.x=\frac{2y+1}{y-3}.$

Here $x≠2x\neq 2$ in the original formula, and the final expression also requires $y≠3y\neq 3$ . In fact, the original formula can never produce $y = 3$ .

When the subject appears more than once

If the required variable appears in several terms, you cannot usually isolate each occurrence separately. Instead:

collect every term containing the variable on one side;
move the remaining terms to the other side;
factorise by taking out the required variable;
divide by the remaining factor.

Key Idea

Collect, factorise, divide

When the new subject appears in more than one term, factorisation turns its several appearances into one common factor that can be isolated.

Example

Collecting repeated occurrences

Make $x$ the subject of

a (x + 2) = b (x - 1) .

Expand both sides so that the terms containing $x$ can be collected:
$a x + 2 a = b x - b .$
Put the $x$ -terms on the same side and the other terms on the opposite side:
$a x - b x = - b - 2 a .$
Factorise the left-hand side:
$x (a - b) = - (b + 2 a) .$
Divide by $a - b$ and simplify the signs:
$x=b+2ab−a.x=\frac{b+2a}{b-a}.$

Common Mistake

Dividing before factorising

From $a x - b x = - b - 2 a$ , do not try to divide the two $x$ -terms separately. First write $x (a - b) = - b - 2 a$ , then divide by the complete coefficient $a - b$ .

Powers and roots

When a power acts on an expression containing the required subject, isolate that powered expression before applying the inverse power.

Example

Undoing a square

Make $r$ the subject of

A=πr2.A=\pi r^2.

Divide by $π\pi$ to isolate $r^2$ :
$r2=Aπ.r^2=\frac{A}{\pi}.$
Taking square roots gives
$r=±Aπ.r=\pm\sqrt{\frac{A}{\pi}}.$
If $r$ represents a radius, the model requires $r≥0r\ge 0$ , so use
$r=Aπ.r=\sqrt{\frac{A}{\pi}}.$

The algebraic equation has both positive and negative solutions, but the meaning of a variable may restrict which values are sensible.

Common Mistake

Forgetting the negative root

Algebraically, if $x^2=k$ with $k > 0$ , then $x=±kx=\pm\sqrt{k}$ . Keep both signs unless the context or a stated restriction rules one out.

Fractional powers

The same method works with other powers. For example, if

V=43πr3,V=\frac{4}{3}\pi r^3,

then isolating $r^3$ and taking a cube root gives

r=3V4π3.r=\sqrt[3]{\frac{3V}{4\pi}}.

A cube root has one real value for every real input, so a $±\pm$ sign is not introduced.

Choosing an efficient order

More than one rearrangement route may be valid. A reliable approach is to look at the operations surrounding the required subject and undo the outermost operation first.

For example, in

y=ax+bc,y=\frac{ax+b}{c},

the expression $a x + b$ is first formed and then divided by $c$ . To recover $x$ , reverse that order: multiply by $c$ , subtract $b$ , then divide by $a$ .

Example

Reversing the order of operations

Make $x$ the subject of

y=4x−53.y=\frac{4x-5}{3}.

Multiply by 3 to undo the division:
$3 y = 4 x - 5.$
Add 5 to undo the subtraction:
$3 y + 5 = 4 x .$
Divide by 4:
$x=3y+54.x=\frac{3y+5}{4}.$

Checking a rearrangement

You can check your result in two useful ways.

Algebraic substitution: substitute your new expression back into the original formula and simplify.
Numerical check: choose permissible values for the variables, evaluate the original formula, and confirm that the rearranged formula returns the starting value.

A numerical check can expose a sign or bracket error, although it is not a proof. Algebraic substitution gives a stronger check.

Tip

Use brackets during substitution

When substituting an expression in place of a variable, put the whole expression in brackets. This prevents powers and multiplication from being applied to only part of it.

Exam technique

In the exam

Identify the variable that must become the subject and note every place where it occurs.
Preserve equality by applying inverse operations to both sides, keeping useful brackets intact where possible.
If the variable appears more than once, collect its terms, factorise and then divide.
State any important restrictions, and use the context to decide whether all algebraic solutions are valid.
Check signs, brackets and denominators by substituting your result into the original formula.

Self review

Check yourself

Can you make $h$ the subject of $V=πr2hV=\pi r^2h$ ?
Can you make $x$ the subject of $p=x+42x−1p=\frac{x+4}{2x-1}$ ?
Why does making $x$ the subject of $y=x^2+3$ usually produce two possible expressions?

What you'll learn

Formulae and subjects

Maintaining equality

Inverse operations

Rearranging formulas with brackets

Formulas involving fractions

A subject in the numerator and denominator

When the subject appears more than once

Powers and roots

Fractional powers

Choosing an efficient order

Checking a rearrangement

Review 1.2.3 Change the subject of a formula by teaching Genie

1.2.3 Change the subject of a formula

What you'll learn

Formulae and subjects

Maintaining equality

Inverse operations

Rearranging formulas with brackets

Formulas involving fractions

A subject in the numerator and denominator

When the subject appears more than once

Powers and roots

Fractional powers

Choosing an efficient order

Checking a rearrangement

Review 1.2.3 Change the subject of a formula by teaching Genie

1.2.3 Change the subject of a formula

What you'll learn

Formulae and subjects

Maintaining equality

Inverse operations

Rearranging formulas with brackets

Formulas involving fractions

A subject in the numerator and denominator

When the subject appears more than once

Powers and roots

Fractional powers

Choosing an efficient order

Checking a rearrangement

Review 1.2.3 Change the subject of a formula by teaching Genie

1.2.3 Change the subject of a formula Revision Guide

1.2.3 Change the subject of a formula

What you'll learn

Formulae and subjects

Maintaining equality

Inverse operations

Rearranging formulas with brackets

Formulas involving fractions

A subject in the numerator and denominator

When the subject appears more than once

Powers and roots

Fractional powers

Choosing an efficient order

Checking a rearrangement

Review 1.2.3 Change the subject of a formula by teaching Genie

1.2.3 Change the subject of a formula Revision Guide