GCSE Maths

Numbers
37 Topics | 1 Quiz
Integers
Prime Numbers and Composite Numbers
Negative Numbers
Comparing Numbers Using Symbols
The Square Numbers
The Laws of Indices
Negative Indices
The Cube Numbers
Triangular Numbers
The Fibonacci Sequence
Factors
Multiples
The Highest Common Factor
Fractions
Adding and Subtracting Fractions
Multiplication of Fractions
Division of Fractions
Fractional Indices
Decimal Places
Fractions and Decimals
1 of 2
Vectors
1 Topic
Vectors
Probability
6 Topics
Introduction to probability
Finding the sum of probabilities
Conditional probability
Set Notation
Venn diagrams
The Complement of a Set
Statistics
2 Topics
Population and samples
Representing data
Algebra
16 Topics | 1 Quiz
Algebraic Notation
Algebraic Vocabulary
Collecting Like Terms
Expanding Brackets and Simplifying
Expanding Double Brackets
The Difference of Two Squares
Substituting into Formulae
Subject of Formula
Factorisation
Factorising Quadratic Expressions
More on Factorisation
Equations
Quadratic Equations
Mathematical Formulae
Algebraic Equivalence and Proof
Functions
Algebra
Sequences
2 Topics | 1 Quiz
Generating sequences
Nth Term Sequences
Sequences
Ratio, Proportion and Rates of Change
8 Topics | 1 Quiz
Inverse Proportion
Conversion
Scale Factors
Percentages
Direct Proportion
Rate, Ratio and Proportion Examples
Ratios
Ratios
Ratio, Proportion and Rates of Change
Geometry and Measures
9 Topics
Loci and constructions
Properties of angles
Properties of polygons
Transformations
Enlargement
Circle theorems
Plane figure properties
Plans and elevations
Areas and Volumes in Map Scale Problems
Mensuration and Calculation
10 Topics
Map scales
Bearings
Area
Volume
Circles: arcs and sectors
Congruence and similarity
Pythagoras’ theorem
Sine and cosine rule
Area of a triangle
Standard units of measure
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Subject of Formula

GCSE Maths Algebra Subject of Formula

Consider the expression y=mx+c. On the left hand side, we only have y. So, we say that y is the subject of formula.

Similarly, in the formula: S = ut + \frac{1}{2} at^{2}, S is the subject of formula.

  • It will be helpful to express a given equation in a certain desired form.

In y=2x+z, y is the subject of formula.

We can rearrange y=zx+c to make z the subject of formula:

y=2x+z

2x+z=y

Subtracting 2x on both sides, we have:

z=y-2x

Now z is the subject of formula.

We can also make x the subject of formula.

y=2x+z

2x+z=y

Subtracting z on both sides, it becomes:

2x=y-z

By dividing both sides by 2, we have:

x= \frac{y-z}{2} or \frac{1}{2}(y-z)

The expressions below are all the same, but they look different.

  • y=2x+z
  • z=y-2x
  • x= \frac{y-z}{2} or \frac{1}{2}(y-z)

Let’s look at some examples.

Example

Make x the subject of formula if x+2y^{2}=5w

x+2y^{2}=5w

Subtract 2y^{2} on both sides

x=5w-2y^{2}

Example

Again, consider x + 2y^{2}=5w. Make y the subject of formula.

x+2y^{2}=5w

Subtract x on both sides

2y^{2}=5w-x

Now divide by 2 on both sides

y^{2} = \frac{5w-x}{2}

We are not quite finished, because we still need to get y.

We take the square root of both sides

y= \sqrt{\frac{5w-x}{2}}

Example

Make w the subject of formula in 2x + ay = y-w

2x+ay=y-w

Add w on both sides

2x+ay+w=y

Now subtract 2x and ay on both sides

w=y-2x-ay

Example

Make y the subject of formula if \sqrt{x+y}=z

\sqrt{x+y}=z

We first remove the square root by squaring both sides

x+y=z^{2}

Now subtract x on both sides

y=z^{2}-x

Example

Make w the subject of formula from the following:

i) \frac{1}{w}=x+y

ii) 3+aw=bw+y

iii) 2x-w=aq+b

i) \frac{1}{w}=x+y

Multiply by w on both sides

1=w(x+y)

Re-write this as w(x+y)=1

Now, divide both sides by (x+y)

w = \frac{1}{x+y}

ii) 3+aq=bw+y

aw-bw=y-3

Take w as a common factor

w(a-b)=y-3

w = \frac{y-3}{a-b}

iii) 2x - w = aw + b

2x-b=aw+w

2x-b=w(a+1)

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

Example

The formula for the area of a trapezium is:

A = \frac{1}{2}h(a+b)

Find an expression for a in terms of A, h and b.

A = \frac{1}{2}h(a+b

2A = h(a+b)

a+b = \frac{2A}{h}

a = \frac{2A}{h} - b

Example

Make y the subject of formula if \sqrt{\frac{x+z}{y}}=w-a

\sqrt{\frac{x+z}{y}}=w-a

\frac{x+z}{y}=w-a=(w-a)^{2}

y = (x+z)(w-a)^{2}

Example

In v^{2} = u^{2}+2ax, make x the subject of formula.

V^{2} = u^{2} + 2ax

2ax = V^{2} - u^{2}

x = \frac{V^{2} - u^{2}}{2a}

Example

Make y the subject of formula if x - y^{3} = w-a

x - y^{3} = w-a

y^{3} = x-w + a

Take the cube root of both sides

y = (x - w +a)^{\frac{1}{3}}

or

y=\sqrt[3] {x-w+a}

Example

Make w the subject of formula if y=\sqrt[3] {x-w+a}

\frac{a+t}{w} = \frac{w}{b}

b(a+t) = w^{2}

w^{2} = b(a + t)

w = \sqrt{b(a + t)}

Example

Given that d = \sqrt{7e - 4f}

a) Find d when e=3 and f=-1

b) Re-arrange the formula to make f the subject.

d = \sqrt{7e - 4f}

a) Subtracting e=3 and f=-1. so:

d = \sqrt{7(3) - 4(-1)}

d = \sqrt{21 + 4}

d = \sqrt{25}

d = 5

b) d = \sqrt{7e-4f}

Squaring both sides

d^{2} = 7e-4f

Transposing 4f to the left and d^{2} to the right:

4f = 7e - d^{2}

Divide both sides by 4

f = \frac{7e - d^{2}}{4}

Or we may right:

f = \frac{1}{4}(7e - d^{2})

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