Constructing and rearranging simple formulas — 0580
Reverse a linear formula in which the subject appears once.
Core & Extended · Syllabus C2.1, C2.5, E2.1, E2.5
Notes and key methods
A formula describes a relationship for a range of inputs; the subject is the variable isolated on its left. Reverse the operations in their opposite order. If \(\displaystyle y=ax+b\) with \(\displaystyle a\ne 0\), subtract b before dividing by a.
Use brackets around the whole numerator when dividing. Units can reveal a misplaced addition or scale factor. Defining letters and constructing the original relationship is as important as rearranging it.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Constructing and rearranging simple formulas
The relationship is \(y=2x+7\). Capital A and B, where used, denote coefficients in a rearranged rule.
Make x the subject of the formula.
Find x when \(\displaystyle y=23\).
Write x=Ay+B. Find A.
Write x=Ay+B. Find B.
Show the reasoning
(a)
Subtract 7 from both sides and divide the entire result by 2.
Answer: \(\displaystyle x=\frac{y-7}{2}\)
(b)
Subtract the additive constant, then divide the whole result by the nonzero multiplier.
\(8\)
Answer: \(8\).
(c)
Subtract the additive constant, then divide the whole result by the nonzero multiplier.
\(\displaystyle \frac{1}{2}\)
Answer: \(\displaystyle \frac{1}{2}\).
(d)
Subtract the additive constant, then divide the whole result by the nonzero multiplier.
\(\displaystyle - \frac{7}{2}\)
Answer: \(\displaystyle - \frac{7}{2}\).
Example 2 · Constructing and rearranging simple formulas
The relationship is \(y=3x+8\). Capital A and B, where used, denote coefficients in a rearranged rule.
Make x the subject of the formula.
Find x when \(\displaystyle y=29\).
Write x=Ay+B. Find A.
Write x=Ay+B. Find B.
Show the reasoning
(a)
Subtract 8 from both sides and divide the entire result by 3.
Answer: \(\displaystyle x=\frac{y-8}{3}\)
(b)
Subtract the additive constant, then divide the whole result by the nonzero multiplier.
\(7\)
Answer: \(7\).
(c)
Subtract the additive constant, then divide the whole result by the nonzero multiplier.
\(\displaystyle \frac{1}{3}\)
Answer: \(\displaystyle \frac{1}{3}\).
(d)
Subtract the additive constant, then divide the whole result by the nonzero multiplier.
\(\displaystyle - \frac{8}{3}\)
Answer: \(\displaystyle - \frac{8}{3}\).
Example 3 · Constructing and rearranging simple formulas
The relationship is \(y=4x+9\). Capital A and B, where used, denote coefficients in a rearranged rule.
Make x the subject of the formula.
Find x when \(\displaystyle y=45\).
Write x=Ay+B. Find A.
Write x=Ay+B. Find B.
Show the reasoning
(a)
Subtract 9 from both sides and divide the entire result by 4.
Answer: \(\displaystyle x=\frac{y-9}{4}\)
(b)
Subtract the additive constant, then divide the whole result by the nonzero multiplier.
\(9\)
Answer: \(9\).
(c)
Subtract the additive constant, then divide the whole result by the nonzero multiplier.
\(\displaystyle \frac{1}{4}\)
Answer: \(\displaystyle \frac{1}{4}\).
(d)
Subtract the additive constant, then divide the whole result by the nonzero multiplier.
\(\displaystyle - \frac{9}{4}\)
Answer: \(\displaystyle - \frac{9}{4}\).
Common mistakes
Keep intermediate values exact until the final step.