Skill: literal-equations
Literal Equations
A formula is an equation with an opinion about which variable matters. Rearrange it, and any variable can be the star.
1 Understand
The core idea in plain language.
A literal equation uses letters as constants: ax + b = c. Solving for x works exactly like numbers — collect x-terms, isolate: ax = c − b, so x = (c − b)/a. The letters a, b, c just ride along like numbers you have not met yet.
To solve y = mx + b for x: subtract b → y − b = mx, divide by m → x = (y − b)/m. Every step is a legal inverse operation; the letters are never an excuse to skip one.
Formulas are meant to be rearranged before use. If you need l from P = 2l + 2w many times, solve once — l = (P − 2w)/2 — then substitute numbers. Rearranging once beats re-solving every time.
Check a rearrangement by substituting numbers: pick values for the letters, compute your expression, and confirm the original equation balances. If it balances for random numbers, the algebra is right.
2 See it
Diagrams that make the idea visual.
Letters ride along like numbers
Solve ax + b = c for x. Treat a, b, c as ordinary numbers.
ax = c − b → x = (c − b)/a.
Rearrange once, use many times
Need the length l from P = 2l + 2w for five different rectangles? Solve for l once: l = (P − 2w)/2. Then each rectangle is one substitution.
With P = 30, w = 4: l = (30 − 8)/2 = 11.
3 The key rule
1) Decide the target variable. 2) Collect its terms on one side (distribute first if needed). 3) Factor/isolate: divide by its coefficient — a letter is fine. 4) Verify with a numeric substitution.
4 Worked examples
Follow each step. The pattern is always the same.
- Subtract b from both sides: ax = c − b.
- Divide by a: x = (c − b)/a.
Check: Try a = 2, b = 3, c = 11: x = (11−3)/2 = 4, and 2(4)+3 = 11. Correct.
- Subtract b: y − b = mx.
- Divide by m: x = (y − b)/m.
Check: Try m = 2, b = 1, y = 9: x = (9−1)/2 = 4, and 2(4)+1 = 9. Correct.
- Subtract 2w: P − 2w = 2l.
- Divide by 2: l = (P − 2w)/2.
Check: Try P = 30, w = 4: l = (30−8)/2 = 11, and 2(11)+2(4) = 30. Correct.
- Substitute r = 60, t = 2.5: d = 60 × 2.5.
- d = 150.
Check: 150/60 = 2.5. Correct.
5 Common mistakes
These errors show up on almost every quiz. Spot them now and they will never cost you points.
ax + b = c → “x + b = c/a” — the b escaped the division.
Divide every term: x = (c − b)/a. The parentheses are the whole point.
“Solve y = mx + b for x” → “y − b = mx” and stop — that isolates mx, not x.
Finish the job: x = (y − b)/m. The target must stand completely alone.
“l = P/2 − 2w looks right.” (It is not: 2w was never divided by 2.)
Substitute P = 30, w = 4: correct l = 11, wrong formula gives 7. The check catches it.
6 Key vocabulary
Say these words like you mean them.
- Literal equation
- An equation with two or more variables; letters act as constants.
- Formula
- A literal equation expressing a relationship, e.g. P = 2l + 2w.
- Solve for
- Rearrange so the named variable stands alone on one side.
- Substitute
- Replace variables with numbers to evaluate or verify.
7 Quick check
Try each one on paper first, then reveal the answer.
Divide by l: w = A/l.
Divide by lw: h = V/(lw).
Add by: ax = c + by; divide by a: x = (c+by)/a.
F = 4 × 5 = 20.
Key points to remember
- Treat the other letters as numbers you have not met — the same inverse operations apply.
- Divide every term on the side; keep the parentheses until the end.
- Rearrange once, then substitute numbers as often as you like.
- Verify every rearrangement with one numeric substitution.