Changing the subject of a formula

Key points

An image of a circle with radius marked r. Written above: A equals area of a circle, r equals radius. Written below: A equals pi r squared, r equals question mark. — Image caption,
Rearranging the formula for the circumference of a circle makes the radius the subject.

A is a rule that links and is used to work something out.
By manipulating the algebra, a different variable is to become the . This is called to change the subject.
By , the subject of a formula is written first, before the equals sign, although it is still correct written on either side.
Manipulating the formula uses skills of understanding algebraic notation and , as well as function machines. The same skills used here can be used when solving equations.

Rearrange a formula using inverse operations

are used to undo each process step in a to change the subject.

To change the subject of a formula:

Write the formula and identify the that is to be the new subject.
Start with the new subject and write out the function machine for the formula, showing each operation step by step.
Undo each step by using the inverse operation:
- Start with the last operation and work back to the first.
- For each step make sure that the inverse is applied to both sides of the formula.
When the new subject is isolated, the rearranging process is complete.
Write the new subject equal to the .

Examples

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A formula is rearranged by using inverse operations. Inverse operations include addition and subtraction, multiplication and division, and also square and square root.
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To undo an operation, the inverse operation is used with the same amount. The inverse of + 5 is – 5. The inverse of × 𝒏 is ÷ 𝒏.
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Rearrange the formula for the area of a square to make the side length the subject.
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The function machine is a one-step process, square. The formula is rearranged by using the inverse of squaring which is square root. Square root both sides of the formula. This isolates the side length, 𝐬. With the side length as the subject of the formula it is 𝐬 = √𝑨
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Rearrange the formula for the area of a triangle to make the height the subject.
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The function machine is a 2-step process: Multiply, by the base, 𝒃, and then divide by two. To rearrange the formula the steps are undone using the inverse operations. The inverse of divide by 2 is multiply by 2, the inverse of multiply by 𝒃 is divide by 𝒃.
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Start with the last operation, divide by 2. Undo this step by using the inverse operation, multiply by 2 on both sides of the formula. The formula is now 2𝑨 = 𝒃𝒉.
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The next operation to undo is multiply by 𝒃. Use the inverse operation, divide by 𝒃 on both sides of the formula. The formula is now 2𝑨/𝒃 = 𝒉 . The 𝒉 has been isolated, the rearranging process is complete. With the height as the subject the formula is 𝒉 = 2𝑨/𝒃.
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Rearrange the formula for the circumference of a circle to make the radius the subject.
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The function machine is a 2-step process: Multiply by two and then multiply by π. These two operations can be treated as a single operation, multiply by 2π. To rearrange the formula this step is undone by using the inverse operation. The inverse of multiplying by 2π is to divide by 2π. Divide both sides of the formula by 2π. This isolates the radius, 𝒓. With the radius as the subject of the formula it is 𝒓 = 𝑪/2π