2 Rearranging a formula

In the formula for the area of a circle, A = π r 2 , we say that A is the subject of the formula. A variable is the subject of the formula if it appears by itself on one side of the formula, usually the left-hand side, and nowhere else in the formula . If we are asked to transpose the formula for r , or solve for r , then we have to make r the subject of the formula. When transposing a formula whatever is done to one side is done to the other . There are five rules that must be adhered to.

Key Point 22
Rearranging a formula

You may carry out the following operations

add the same quantity to both sides of the formula

subtract the same quantity from both sides of the formula

multiply both sides of the formula by the same quantity

divide both sides of the formula by the same quantity

take a ‘function’ of both sides of the formula: for example,

  find the reciprocal of both sides (i.e. invert).

Example 64

Transpose the formula p = 5 t 17 for t .

Solution

We must obtain t on its own on the left-hand side. We do this in stages by using one or more of the five rules in Key Point 22. For example, by adding 17 to both sides of p = 5 t 17 we find

p + 17 = 5 t 17 + 17 so that p + 17 = 5 t

Dividing both sides by 5 we obtain t on its own:

p + 17 5 = t

so that t = p + 17 5 .

Example 65

Transpose the formula 2 q = p for q .

Solution

First we square both sides to remove the square root. Note that ( 2 q ) 2 = 2 q . This gives

2 q = p 2

Second we divide both sides by 2 to get q = p 2 2 .

Note that in general by squaring both sides of an equation may introduce extra solutions not valid for the original equation. In Example 65 if p = 2 then q = 2 is the only solution. However, if we transform to q = p 2 2 , then if q = 2 , p can be + 2 or 2 .

Task!

Transpose the formula v = t 2 + w for w .

You must obtain w on its own on the left-hand side. Do this in several stages.

First square both sides to remove the square root:

v 2 = t 2 + w

Then, subtract t 2 from both sides to obtain an expression for w :

v 2 t 2 = w

Finally, write down the formula for w :

w = v 2 t 2

Example 66

Transpose x = 1 y for y .

Solution

We must try to obtain an expression for y . Multiplying both sides by y has the effect of removing this fraction:

 Multiply both sides of  x = 1 y  by  y  to get  y x = y × 1 y  so that  y x = 1

Divide both sides by x to leaves y on its own, y = 1 x .

Alternatively: simply invert both sides of the equation x = 1 y to get 1 x = y .

Example 67

Make R the subject of the formula

2 R = 3 x + y

Solution

In the given form R appears in a fraction. Inverting both sides gives

R 2 = x + y 3

Thus multiplying both sides by 2 gives

R = 2 ( x + y ) 3

Task!

Make R the subject of the formula 1 R = 1 R 1 + 1 R 2 .

  1. Add the two terms on the right:

    1 R 1 + 1 R 2 = R 2 + R 1 R 1 R 2

  2. Write down the complete formula:

    1 R = R 2 + R 1 R 1 R 2

  3. Now invert both sides:

    R = R 1 R 2 R 2 + R 1