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Expanding brackets Factorising Solving equationsThis topic is relevant for:

Here we will learn about **making x 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

For example,

Make

**Step-by-step guide:** Rearranging equations

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. factorise2x + 3xy x(2+3y) **not always required** - Perform an operation to ensure only a single
x variable is left as the subject.

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.

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**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

\[x=6y+5\]

This is **wrong **because we have only multiplied the

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

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 as an inverse operation the answer can be positive or negative.

E.g.

\begin{align}
& {{x}^{2}}=4 \\
& x=\pm \sqrt{4}=\pm 2 \\
\end{align}

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

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+7=5x

**(1)**

\frac{y+7}{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{3t}{y-15} \quad

**(1)**

You have now learned how to:

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

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