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})