Changing the Subject — When the Letter Appears Twice
Grade 7The harder case
Sometimes the letter you want to make the subject appears in more than one place. You cannot just "move it across" — you need to gather all the terms containing that letter on one side, then factorise it out to isolate it.
Worked example — subject appears twice
EXAMPLE
Make x the subject of: ax + 3 = bx + c
Gather x terms on one side: ax − bx = c − 3. Factorise: x(a − b) = c − 3. Divide: x = (c − 3) / (a − b).
Worked example — subject inside a fraction
EXAMPLE
Make t the subject of: s = (t + 3) / (t − 1)
Multiply both sides by (t − 1): s(t − 1) = t + 3. Expand: st − s = t + 3. Gather t terms: st − t = s + 3. Factorise: t(s − 1) = s + 3. Divide: t = (s + 3) / (s − 1).
The key move to remember
Whenever you see the target letter on both sides after expanding, your next move is always: gather → factorise → divide. That three-step sequence handles any rearrangement of this type.
Try this one
The area of a circle is given by \( A = \pi r^2 \).
Make \( r \) the subject of the formula.
[2]
Nova's hint:
Try to Solve it with Nova?
This is rearranging a formula where the wanted letter is squared.
Ask yourself: 'once the squared letter is on its own, what operation undoes squaring?' Remember the opposite of squaring is taking a square root, as in \( V = k n^2 \) leading to \( n = \sqrt{\tfrac{V}{k}} \).
