Subject of Formula

Consider the equation y=mx+c. On the left-hand side, we only have y, so we say that y is the subject of the formula. Similarly, in the formula: S = ut + \frac{1}{2} at^{2}, S is the subject of the formula.

It is often helpful to express a given equation with a specific variable as the subject. Let’s look at an example:

In the equation y=2x+z, y is the subject of the formula. We can rearrange the equation to make z the subject of the formula:

1. Start with the equation y=2x+z.

2. Subtract 2x from both sides: z=y-2x.

Now z is the subject of the formula.

We can also make x the subject of the formula:

1. Start with the equation y=2x+z.

2. Subtract z from both sides: 2x=y-z.

3. Divide both sides by 2: x= \frac{y-z}{2} or \frac{1}{2}(y-z).

The expressions below are all equivalent but look different:

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

Let’s look at more examples that involve changing the subject of a formula.

Examples

Example 1:

Given the equation x+2y^{2}=5w, make x the subject of the formula.

Start with the equation x+2y^{2}=5wx+2y^{2}=5w.

Subtract 2y^{2}2y^{2} from both sides: x=5w-2y^{2}x=5w-2y^{2}.

Example 2:

Given the equation x+2y^{2}=5w, make y the subject of the formula.

1. Start with the equation x+2y^{2}=5wx+2y^{2}=5w.

2. Subtract x from both sides: 2y^{2}=5w-x2y^{2}=5w-x.

3. Divide both sides by 2: y^{2} = \frac{5w-x}{2}y^{2} = \frac{5w-x}{2}.

4. Take the square root of both sides: y= \sqrt{\frac{5w-x}{2}}y= \sqrt{\frac{5w-x}{2}}.

Example 3:

Given the equation 2x+ay=y-w, make w the subject of the formula.

1. Start with the equation 2x+ay=y-w2x+ay=y-w.

2. Add w to both sides: 2x+ay+w=y2x+ay+w=y.

3. Subtract 2x and ay from both sides: w=y-2x-ayw=y-2x-ay.

Example 4:

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

\sqrt{x+y}=z\sqrt{x+y}=z

Square both sides to remove the square root:

x+y=z^{2}x+y=z^{2}

Now subtract x on both sides:

y=z^{2}-xy=z^{2}-x

Example 5:

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

Multiply by w on both sides:

1=w(x+y)1=w(x+y)

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

Now, divide both sides by (x+y)(x+y):

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

ii) 3+aw=bw+y3+aw=bw+y

aw-bw=y-3aw-bw=y-3

Take w as a common factor:

w(a-b)=y-3w(a-b)=y-3

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

iii) 2x - w = aw + b2x - w = aw + b

2x-b=aw+w2x-b=aw+w

Factor out w:

w(1+a)=2x-bw(1+a)=2x-b

Divide by (1-a)(1-a):

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

Example 6:

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

2A = h(a+b)2A = h(a+b)

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

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

Example 7:

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

x - y^{3} = w-ax - y^{3} = w-a

Add y^{3}y^{3} and aa to both sides, and add w:

y^{3} = x - w + ay^{3} = x - w + a

Take the cube root of both sides:

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

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