2 Rearranging a formula
In the formula for the area of a circle, , we say that 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 , or solve for , then we have to make 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
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 for .
Solution
We must obtain 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 we find
Dividing both sides by 5 we obtain on its own:
so that .
Example 65
Transpose the formula for .
Solution
First we square both sides to remove the square root. Note that . This gives
Second we divide both sides by to get .
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 then is the only solution. However, if we transform to , then if , can be or .
Task!
Transpose the formula for .
You must obtain on its own on the left-hand side. Do this in several stages.
First square both sides to remove the square root:
Then, subtract from both sides to obtain an expression for :
Finally, write down the formula for :
Example 66
Transpose for .
Solution
We must try to obtain an expression for . Multiplying both sides by has the effect of removing this fraction:
Divide both sides by to leaves on its own, .
Alternatively: simply invert both sides of the equation to get .
Example 67
Make the subject of the formula
Solution
In the given form appears in a fraction. Inverting both sides gives
Thus multiplying both sides by 2 gives
Task!
Make the subject of the formula .
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Add the two terms on the right:
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Write down the complete formula:
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Now invert both sides: