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GCSE Maths Algebra Algebraic Expressions

Rearranging Equations

Rearranging Equations

Here is everything you need to know about rearranging equations for GCSE maths (Edexcel, AQA and OCR). You’ll learn what rearranging equations means and how to change the subject of the formula.

Look out for the rearranging equations worksheets and exam questions at the end.

What is rearranging equations?

Rearranging equations changes the form of the equation to display it in a different way. This is sometimes called changing the subject.

How to rearrange equations step by step:

  1. Identify the variable you need to make the subject.
  2. Isolate the variable by: removing any fractions by multiplying by the denominator/s, dividing by the coefficient of the variable, and by square rooting or squaring both sides of the equation
  3. Perform inverse operations so that each term that needs to be made the subject is on one side of the equation.

What does rearranging equations mean?

When we rearrange an equation we change the form of the equation to display it in a different way.

For example, the below three equations are rearranged forms of exactly the same equation.

\[\begin{aligned} a-b &=2 \\ a &=b+2 \\ a-2 &=b \end{aligned}\]

Typically we rearrange equations and formulas by using inverse operations to make one variable the subject of the formula. The subject of the formula is the single variable that is equal to everything else. i.e. the term by itself on one side of the equal sign.


Here are some example where s is the subject of the formula

\[s=4+a \qquad\qquad s=5 f-6 c+8 \qquad s=\frac{5+t}{y-8}\]

To do this we move variables and constants (numbers) to the other side of the equation from the variable we are trying to make the subject of the formula.

What do we mean by rearranging equations?

What do we mean by rearranging equations?

What do we mean by rearranging equations?

What do we mean by rearranging equations?

Check out our main algebraic expressions lesson for a complete summary of algebraic expressions, and then explore our other lessons for detailed step-by-step guides and worksheets on each type.

Rearranging Equations worksheets

Rearranging Equations worksheets

Rearranging Equations worksheets

Get your free rearranging equations worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Rearranging Equations worksheets

Rearranging Equations worksheets

Rearranging Equations worksheets

Get your free rearranging equations worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

How to rearrange formula to change the subject of the formula

In order to do rearrange formula to change the subject of the formula, I need to follow the steps:

  1. Identify the variable you need to make the subject of the formula.
  2. Isolate the variable – this step may look slightly different depending on the format of the question.
    – Remove any fractions by multiplying by the denominator/s
    – Divide by the coefficient of the variable
    – Square root or square both sides of the equation
    *not always required*
  3. Rearrange the equation so each term containing the term you want to be the subject is on one side of the equation – normally the left-hand side.
  4. Factorisation may be needed if you have multiple different terms containing your subject e.g. factorise 2x + 3xy to x(2 + 3y)
    *not always required*
  5. Perform an operation to ensure only the single variable is left as the subject.

How to rearrange formula to change the subject of the formula

How to rearrange formula to change the subject of the formula

Rearranging equations examples

Example 1: multiple step but with single variable

p = 2(x − 3)

  1. Identify the variable to be made the subject.

\[p = 2(x − 3)\]

In this question it is x.

2 Divide each side of the equation by 2

Rearranging equations step 1

3 Add 3 to each side of the equation

Rearranging equations examples

Answer:

\[\frac{p}{2}+3= x\]

Fully worked out answer:

Rearranging equations worked out example

Example 2: questions involving x2

\[y=x^{2}-4\]

\[y=x^{2}-4\]

In this question it is x

Answer:

\[\pm\sqrt{y+4}=x\]

Fully worked out answer:

Rearranging equations gcse maths help

Example 3: questions involving √x

\[y=\sqrt{3 x}+n\]

\[y=\sqrt{3 x}+n\]

In this question it is x.

Rearranging equations 1

Rearranging equations 1

Rearranging equations examples

Answer:

\[\frac{(y-n)^{2}}{3}=x\]

Fully worked out answer:

Rearranging equations gcse

Example 4: factorisation of the variable is required

\[y=\frac{2 x z}{x-5}\]

\[y=\frac{2 x z}{x-5}\]

In this question it is x.

Answer:

\[x =\frac{5 y}{y-2 z}\]

Fully worked out answer:

Example 5: factorisation of the variable is required

\[\frac{a}{3}=\frac{2-7 x}{x-5}\]

\[\frac{a}{3}=\frac{2-7 x}{x-5}\]

In this question it is x.

rearranging equations examples gcse

rearranging equations examples gcse

rearranging equations examples gcse

rearranging equations examples gcse

Answer:

\[x =\frac{6+5 a}{a+21}\]

Fully worked out answer:

Rearranging equations worked out examples

Common misconceptions

  • When we perform an operation to one side of the equation we have to do to the other.
  • Incorrect use of the inverse operation.
  • Incorrectly following the order of operations.
  • All variables of the subject need to be on one side of the equal sign.
\[E.g. x = 2x+2\]

  • When we square root a number/variable the answer can be positive or negative.
\[E.g. \sqrt{4}=\pm{2}\]
√x should be written as ± √x

  • Not factorising when we have the subject in more than one term.
E.g. Make x the subject
\[\begin{aligned} 2x+3xy&=3y\\ x(2+3y)&=3y\\ x&=\frac{3y}{2+3y} \end{aligned}\]

Practice rearranging equations questions

1. Make a the subject of the formula h=3(a+7)

h -7=a
GCSE Quiz False

\frac{h}{3} -7=a
GCSE Quiz True

\frac{h}{3} -21=a
GCSE Quiz False

h-21=a
GCSE Quiz False
h=3(a+7)

 

Divide both sides by 3

 

\frac{h}{3}=a+7

 

Subtract 7 from both sides

 

\frac{h}{3}-7=a

2. Make b the subject of the formula p=b^{2}-9 k

p+9k=b
GCSE Quiz False

\pm \sqrt{p+9 k}=b
GCSE Quiz True

\pm \sqrt{p-9 k}=b
GCSE Quiz False

\pm \sqrt{9k-p}=b
GCSE Quiz False
p=b^{2}-9k

 

Add 9k to both sides

 

p+9k=b^{2}

 

Square root both sides

 

\pm \sqrt{p+9 k}=b

3. Make c the subject of the formula g=\sqrt{5 c-r}

g^{2}+r=c
GCSE Quiz False

\frac{g^{2}-r}{5}=c
GCSE Quiz False

\frac{g^{2}+r}{5}=c
GCSE Quiz True

g^{2}-r=c
GCSE Quiz False
g=\sqrt{5 c-r}

 

Square both sides

 

g^{2}=5c-r

 

Add r to both sides

 

g^{2}+r=5c

 

Divide both sides by 5

 

\frac{g^{2}+r}{5}=c

4. Make d the subject of the formula y=\frac{3d+1}{4d}

d=\frac{1}{4y}-3
GCSE Quiz False

d=\frac{1}{4y-3}
GCSE Quiz True

d=\frac{1}{4y+3}
GCSE Quiz False

d=\frac{1}{4y}+3
GCSE Quiz False
y=\frac{3d+1}{4d}

 

Multiply both sides by 4d

 

4dy=3d+1

 

Subtract 3d from both sides

 

4dy-3d=1

 

Factorise the left hand side

 

d(4y-3)=1

 

Divide both sides by 4y-3

 

d=\frac{1}{4y-3}

5. Make e the subject of the formula \frac{q}{3} = \frac{6-2e}{e+1}

e=\frac{18-q}{q-6}
GCSE Quiz False

e=\frac{18-q}{q+6}
GCSE Quiz True

e=\frac{18+q}{q+6}
GCSE Quiz False

e=\frac{18-q}{6}
GCSE Quiz False
\frac{q}{3} = \frac{6-2e}{e+1}

 

Multiply both sides by 3(e+1)

 

qe+q=18-6e

 

Subtract q , and add 6e to both sides

 

qe+6e=18-q

 

Factorise the left hand side

 

e(q+6)=18-q

 

Divide both sides by q+6

 

e=\frac{18-q}{q+6}

6. Make f the subject of the formula \frac{l^{3}}{5} = \frac{4l-3f}{2f+9}

f=\frac{20 l+9 l^{3}}{2 l^{3}+15}
GCSE Quiz False

f=\frac{20 l-9 l^{3}}{2 l^{3}+15}
GCSE Quiz True

f=\frac{20 l-9 l^{3}}{2 l^{3}-15}
GCSE Quiz False

f=\frac{20 l+9 l^{3}}{2 l^{3}-15}
GCSE Quiz False
\frac{l^{3}}{5} = \frac{4l-3f}{2f+9}

 

Multiply both sides by 5(2f+9)

 

2fl^{3}+9l^{3}=20l-15f

 

Add 15f , and subtract 9l^3 , from both sides

 

2fl^{3}+15f=20l-9l^{3}

 

Factorise the left hand side

 

f(2l^{3}+15)=20l-9l^{3}

 

Divide both sides by 2l^3+15

 

f=\frac{20 l-9 l^{3}}{2 l^{3}+15}

Rearranging equations GCSE questions

1. Make x the subject of the formula

\[y=2x+4\]

Show answer

\[y-4=2x\]

(− 4) to make 2x the subject (1)

\[\frac{y-4}{2}=x\]

(÷ 2) to make x the subject (1)

(2 marks)

2. Make s the subject of

\[v^{2}=u^{2}+2as\]

Show answer

\[v^{2}-u^{2}=2as\]

(− u2) to make 2as the subject (1)

\[\frac{v^{2}-u^{2}}{2a}=s\]

(÷ 2a) to make s the subject (1)

(2 marks)

3. Make g the subject of the formula

\[y=\sqrt{\frac{g+6}{5}}\]

Show answer

\[y^{2}=\frac{g+6}{5}\]

Square each side or ‘y2‘ seen (1)

\[5y^{2}=g+6\]

Multiply by denominator, ‘x5′ (1)

\[5y^{2}-6=g\]

Subtract 6 from both sides (1)

(3 marks)

Learning checklist

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

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