Rearranging a formula is the same moves you already know, with letters instead of numbers.
Make h the subject
A = ½bh
Clear the half by multiplying both sides by 2: 2A = bh. Then divide by b: h = 2Ab. B divides by 2 instead of multiplying — undoing a half means doubling, not halving again.
If you can solve 12 = ½(4)h, you can solve A = ½bh.
12 = ½(4)h
h = 24 ÷ 4 = 6
A = ½bh
h = 2A ÷ b
Identical moves: clear the fraction, then divide by whatever multiplies the target.
The answer looks unfinished because it still has letters in it. That is normal — a rearranged formula is the answer.
A = ½bh
the formula
2A = bh
× 2 both sides
h = 2Ab
÷ b both sides
Doubling A means doubling everything on the other side too. Multiplying only part of a side is the most common wrecking move here.
F is wrapped in a bracket and multiplied by a fraction. Unwrap from the outside.
C = (59)(F − 32)
the formula
(95)C = F − 32
× 95 both sides
F = (95)C + 32
+ 32 both sides
Multiplying by 95 is the same as dividing by 59 — flip and multiply.
Sanity check with a known pair: C = 100 gives F = 180 + 32 = 212. Boiling point, correct.
Get it out of the denominator before anything else.
S = a ÷ (1 − r)
r is trapped below
S(1 − r) = a
× (1 − r) both sides
S − Sr = a
distribute
S − a = Sr
gather the r term
r = S − aS
÷ S
Collect the terms, then factor it out.
ax + 3 = bx + 7
x on both sides
ax − bx = 4
gather the x terms
x(a − b) = 4
factor x out
x = 4a − b
÷ (a − b)
You cannot divide by x while x still sits on both sides. Gather first, factor, then divide by the whole bracket.
Solve E = ½mv² for v, where v > 0.
2E = mv²
× 2 both sides
v² = 2Em
÷ m
v = √2Em
square root last; v > 0 keeps only +
Different: the square is the last thing undone, after everything around it has been cleared.
Solve T = 2π√Lg for L.
T2π = √Lg
isolate the root first
T²4π² = Lg
square both sides — the 2π too
L = gT²4π²
× g
Different: a root is undone by squaring, and squaring hits every factor on that side.
Solve y = 7 − 2x for x.
2x + y = 7
+ 2x: make the x term positive
2x = 7 − y
− y
x = 7 − y2
check: y = 1 gives x = 3, and 7 − 6 = 1 ✓
Different: moving the x term across first avoids dividing by −2.
Subtracting 7 and then dividing by 2 instead of −2 gives x = y − 72. Put y = 1 in: it gives −3, not 3.
Lens formula: solve 1f = 1u + 1v for v. Isolate the reciprocal first, combine into one fraction, and only then flip.
1v = 1f − 1u
get 1v alone
1v = u − ffu
common denominator fu
v = fuu − f
flip both sides — one fraction each
From 1v = 1f − 1u it is tempting to write v = f − u. Reciprocals do not spread over + or −: flip only when each side is a single fraction.
Target on top and bottom: solve y = 2x − 1x + 4 for x. Clear the denominator, then it is Step 5 again.
y(x + 4) = 2x − 1
× (x + 4) both sides
xy + 4y = 2x − 1
distribute
xy − 2x = −4y − 1
x terms left, the rest right
x(y − 2) = −(4y + 1)
factor x out
x = 4y + 12 − y
÷ (y − 2), sign moved into the bottom
Check with x = 1: y = 15, and 45 + 12 − 15 = 9595 = 1 ✓. Letters as denominators work the same way: for xp − xq = 1, multiply every term by pq to get qx − px = pq, so x = pqq − p.
“Solve for …” items list four rearrangements that differ by one slip — a flipped sign, a term not multiplied, a reciprocal split. Rather than redo the algebra, put small numbers into the original, then into each choice: the one that gives back the same value is the answer.
Pick easy values, run the original, then run your rearrangement.
Make w the subject
P = 2l + 2w
Subtract 2l first: P − 2l = 2w. Then divide the whole side by 2: w = P − 2l2. C divides only the P — whatever you do has to reach every term on that side.
