One to one maths interventions built for KS4 success

Weekly online one to one GCSE maths revision lessons now available

In order to access this I need to be confident with:

Expanding brackets

Factorising expressions

Solving equations

This topic is relevant for:

Here we will learn about making ** the subject** of an equation or a formula.

There are also rearranging equations worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Making x the subject **of a formula or equation means rearranging the equation or formula so that we have a single

E.g.

Make

In order to make

- Isolate the variable by:

– Removing any fractions by multiplying by the denominator(s).

– Dividing by the coefficient of the variable.

– Adding or subtracting terms near to the variable.

– Taking a root or power of both sides of the equation. - Rearrange the equation so each term containing
x is on the same side of the =. - Factorisation may be needed if there are multiple terms containing
x .

E.g. factorise 2x *+*3xy x (2+3y )**not always required** - Perform an operation to ensure only a single
x variable is left as the subject.

**Read more: **Rearranging equations

Get your free make x the subject worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free make x the subject worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEMake

Make

**Step 1:**

Divide each side of the equation by 3

**Step 2:**

Subtract a from each side of the equation

**Step 1:**

Subtract t from both sides of the equation

**Step 2:**

Square root each side

*Remember the square root can be + or \;− *

**Step 1:**

Add 9a to both sides of the equation

**Step 2:**

The inverse operation of ‘square root’ is to ‘square’ each side

**Step 3:**

The inverse operation of multiply is divide, so divide both sides by 5

**Step 1:**

Multiply each side of the equation by the denominator

**Step 2:**

Expand the bracket on the left hand side of the equation and rearrange the equation. This will help us to get all terms with x onto one side of the equation

**Step 3:**

Factorise the left side of the equation so we have a single variable x.

**Step 4:** Divide by (a - 5b)

This will leave x as the subject of the equation

**Step 1:**

Multiply each side of the equation by the denominator of the other side.

**Step 2:**

Expand the bracket on the LHS and RHS of the equation and rearrange. This will help to get all terms with x onto one side of the equation

**Step 3:**

Factorise the left side of the equation so that we are left with only one of the variable x.

**Step 4:**

Divide both sides by (b + 8) to leave x as the subject

**Perform the same operation to both sides**

When we perform an operation to the left hand side of the equation we have to perform the same operation to the right hand side.

E.g

\[\frac{x}{2}=3 y+5\]

To isolate the variable *x* we need to multiply both sides by 2.

\[x=6y+5\]

This is **wrong **because we have only multiplied the 3*y *by 2.

The correct answer should be:

\[x=2(3y+5)=6y+10\]

This is correct because we have multiplied everything by 2 using brackets.

**Inverse operations**

E.g.

To isolate the variable

E.g.

\[\frac{x}{5}=5 \div x\]

so the inverse operation is × 5.

E.g.

To isolate the variable

The inverse operation of +5 is −5.

E.g.

To isolate the variable

The inverse operation of −5 is +5.

**Factorising**

To make *x* the subject there can only be one *x* visible at the end. We need to get all x’s on one side of the equal sign and then factorise.

E.g.

\[x = ax+y \]

\[\begin{aligned}
x-a x &=y \\
x(1-a) &=y \\
x &=\frac{y}{1-a}
\end{aligned}\]

**Square rooting a term**

When we square rooting a number/variable the answer can be positive or negative.

E.g.

\[\sqrt{4}=\pm{2}\]

\[\sqrt{x} \text { should be written as } \pm \sqrt{x}\]

1. Make x the subject of the formula

y = 6(x+8)

x = y – \frac{3}{4}

x = \frac{y}{6} + 8

x = \frac{y}{6} – 8

x = y – 8

y = 6(x + 8)

Divide both sides by 6

\frac{y}{6} = x + 8

Then subtract 8 from both sides

x = \frac{y}{6} – 8

2. Make x the subject of the formula

3p={x}^2-4b

x=\pm\sqrt{3b+4p}

x=\pm\sqrt{4b-3p}

x=4b+3p

x=\pm\sqrt{4b+3p}

3p={x}^2-4b

Add 4b to both sides

3p+4b=x^{2}

Square root both sides

x=\pm\sqrt{4b+3p}

3. Make x the subject of the formula

6g=\sqrt{7x-8}

x=\frac{6{g}^2+8}{7}

x=\frac{36{g}^2-8}{7}

x=\frac{36{g}^2+8}{7}

x=\frac{6g+8}{7}

6g=\sqrt{7x-8}

Square both sides

(6g)^{2}=(\sqrt{7x-8})^{2}

36g^{2}=7x-8

Add 8 to both sides

36g^{2}+8=7x

Divide both sides by 7

x=\frac{36{g}^2+8}{7}

4. Make x the subject of the formula

y=\frac{4x-f}{5x}

x=\frac{f}{5y-4}

x=\frac{-f}{5y+4}

x=\frac{f}{5y+4}

x=\frac{-f}{5y-4}

y=\frac{4x-f}{5x}

Multiply both sides by 5x

5xy=4x-f

Subtract 4x from both sides

5xy-4x=-f

Factorise the left hand side

x(5y-4)=-f

Divide both sides by the quantity in the bracket

x=\frac{-f}{5y-4}

5. Make x the subject of the formula

\frac{y}{3}=\frac{6-2x}{x+3}

x=\frac{18+3y}{y+6}

x=\frac{18-3y}{y-6}

x=\frac{18-3y}{y+6}

x=\frac{18+3y}{y-6}

\frac{y}{3}=\frac{6-2x}{x+3}

Multiply each side of the equation by the denominator of the other side.

xy+3y=18-6x

To both sides, add 6x and subtract 3y

xy+6x=18-3y

Factorise the left hand side

x(y+6)=18-3y

Divide by the quantity in the bracket

x=\frac{18-3y}{y+6}

1. Make x the subject of the formula

y=5x-7

**(2 marks)**

Show answer

y+5=5x

**(1)**

\frac{y+5}{5} = x

**(1)**

2. Make x the subject of the formula

z^{2}=x^{2}-5 a y

**(2 marks)**

Show answer

z^{2}+5 a y=x^{2}

**(1)**

\pm\sqrt{z^{2}+5 a y}=x

**(1)**

3. Make x the subject of the formula

y=\frac{3(t+5 x)}{x}

**(4 marks)**

Show answer

y x=3 t+15 x

**(1)**

y x-15 x=3 t

**(1)**

x(y-15)=3 t

**(1)**

x=\frac{3}{y-15} \quad

**(1)**

You have now learned how to:

- Understand and use standard mathematical formulae
- Rearrange formulae to change the subject

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths revision programme.