Equations
Literal Equations and Formula Rearrangement
Making a chosen symbol the subject of a formula, why the target must be isolated as a common factor, and the conditions that come with dividing by a variable.
What this page covers
- Rearrange a formula to make a stated symbol the subject
- Collect a target variable appearing in more than one term
- State the conditions under which a rearrangement is valid
- Check a rearrangement by substitution
What changes and what does not
A literal equation carries letters where a numeric equation carries numbers. The method is identical — apply inverse operations to both sides — but two things become more prominent.
Nothing collapses
3 + 5 becomes 8; a + b stays a + b. Every term must be carried along, so the working is longer and the bookkeeping matters more.
Conditions appear
Dividing by 4 is always safe. Dividing by 1 + rt requires it to be non-zero, and that condition belongs in the answer.
The target symbol is called the subject. Making P the subject means producing an expression of the form P = something with no P in it.
The source's worked example
Making P the subject of A = P + Prt
The source describes the context: A is the amount a sum P grows to at simple interest rate r over time t. Given A, r and t, find P.
P appears in both terms, so it cannot be isolated by transposing. It must be factored out first.
The source labels the key step directly: P is a monomial common factor. Recognising that is the whole of the problem; the rest is one division.
The division requires 1 + rt ≠ 0. In this application r and t are non-negative, so 1 + rt ≥ 1 and the condition holds automatically — but it is worth saying why rather than ignoring it.
Take P = 100, r = 0.05, t = 2. Then A = 100 + 100(0.1) = 110. The rearranged formula gives 1101.1 = 100. Correct.
The general procedure
- Clear fractions and radicals affecting the target, by multiplying through or squaring.
- Expand any brackets containing the target.
- Collect every term containing the target on one side, and everything else on the other.
- Factor the target out of that side.
- Divide by the bracket, and state the condition that it is non-zero.
- Check by substituting numbers into both forms.
Steps 3 and 4 are where the work is. If the target appears only once, they are trivial and the rearrangement is straightforward transposition.
Further worked examples
Target appearing once — make r the subject of A = π r2
The source adds the condition r > 0, which is the right call: algebraically r = ±√A/π, but a radius cannot be negative. Context, not algebra, discards the second root.
Target on both sides — make x the subject of ax + b = cx + d
If a = c the equation reduces to b = d, which is either an identity (if b = d) or has no solution (if not). Dividing by a - c without noticing would silently assume the general case.
Nested target — make y the subject of x + yx - y = k
Take x = 3, k = 2. Then y = 3(1)3 = 1, and 3 + 13 - 1 = 2. Correct.
Conditions are part of the answer
A rearranged formula that omits its conditions is incomplete, and in an engineering context that omission can matter.
| Step | Condition generated | Example |
|---|---|---|
| Dividing by an expression | It must be non-zero | P = A1 + rt needs 1 + rt ≠ 0 |
| Taking an even root | The radicand must be non-negative | r = √A/π needs A ≥ 0 |
| Choosing one sign of a root | Justified by context | r > 0 because a radius is positive |
| Multiplying by a denominator | It must be non-zero | x + yx - y = k needs x ≠ y |
Two of these are easy to lose. Squaring can introduce solutions that were never there; dividing by a variable can lose solutions that were. Both are covered in Radical Equations and Spurious Solutions.
Common mistakes
| Mistake | Correct | Remedy |
|---|---|---|
| Transposing when the target appears twice | Collect and factor first | P + Prt cannot be split by transposition |
| Dividing by a bracket that might vanish | State the condition | Ask when the divisor is zero |
| Taking only the positive root without saying why | ± algebraically; context may discard one | Name the reason, e.g. a radius is positive |
| Multiplying only some terms when clearing a fraction | Every term on both sides | Count terms before and after |
| Leaving the target on the right | Write P = … | Convention: the subject goes on the left |
Frequently asked questions
What is a literal equation?
One whose coefficients are letters rather than numbers. Solving means isolating one chosen symbol in terms of the others — the same process as solving a numeric equation, with the arithmetic left undone.
What if the target appears twice?
Collect every term containing it on one side, factor it out, then divide by the bracket. That factoring step is the whole difficulty; without it the target cannot be isolated.
Do I need to state conditions?
Whenever you divide by something containing a variable, yes. Dividing by 1 + rt requires 1 + rt ≠ 0. In an applied context the quantity is often positive by its meaning, and saying so is enough.
Can I check a rearrangement?
Yes. Put numbers into the original, solve numerically, then check the rearranged version reproduces the same value. It takes a minute and catches sign errors reliably.
Source. Handwritten teaching notes, Week 3, page 8.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.