# Changing The Subject

## Change the subject in formulae in which the new subject appears twice.

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I have made $$x$$ the subject of two formulae below but there are some gaps in my working. Fill in the gaps to make the the methods correct.

 $$ax + bx = c$$ $$x(a + b) = c$$ $$x=$$ $$c$$$$a+$$ $$a(x + 3) = 2x$$ $$ax + 3a = 2x$$ $$ax - 2x = -3a$$ $$x(a - 2) = -3a$$ $$x =$$ $$\frac{-3a}{a - 2}$$ $$x=$$ $$3a$$ $$-$$ $$ax + b = cx + d$$ $$ax - cx =$$ $$- b$$ $$x ($$ $$-$$ $$) =$$ $$- b$$ $$x=$$ $$-$$ $$-$$ $$\frac{5a+x}{x}$$$$= 7b$$ $$5a+x=$$ $$5a=$$ $$-$$ $$5a=$$ ( $$-$$ ) $$x=$$ $$-$$
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## Description of Levels

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Level 1 - Formulas which can be rearranged by adding or subtracting terms from both sides

Example: Make e the subject of the formula d = e - f

Level 2 - Formulas which can be rearranged by multiplying or dividing both sides by a value

Example: Rearrange the formula n = mp

Level 3 - Formulas which can be rearranged by adding, subtracting, multiplying or dividing both sides by a value

Example: Rearrange the formula b = a + cd

Level 4 - Formulas including brackets or expressions in the numerator or denominator of a fraction

Example: Rearrange the formula p = s(t + 2)

Level 5 - Formulas including squares or square roots

Example: Rearrange the formula d² = 2a + 1

Level 6 - Finding the unknown which is not the subject of a formula

Example: If m = n² + 2p, find p when m=8 and n=10

Level 7 - Rearrange the formulae where the new subject appears twice; fill in the blanks

Example: Rearrange the formula ax + b = cx + g to make x the subject

Level 8 - Rearrange the formulae where the new subject appears twice; show your working

Example: Rearrange the formula a(3-x)=5x to make x the subject

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## Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

## Example

Make $$x$$ the subject of the following formula:

$$a = \frac{b-x}{2x+c}$$

Multiply both sides of the equation by $$2x+c$$

$$2ax + ac = b-x$$

Add $$x$$ and subtract $$ac$$ from both sides

$$2ax + x = b - ac$$

Factorise the left side

$$x(2a + 1) = b - ac$$

Divide both sides by $$2a + 1$$

$$x= \frac{b - ac}{2a + 1}$$

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