Equations
Conditional and Identical Equations
The difference between an equation true for some values and one true for all of them, why only the first kind is worth solving, and what a solution set actually is.
What this page covers
- Distinguish a conditional equation from an identity
- Write a solution set in set notation
- Recognise an equation with no solution and one with infinitely many
- Say which operations preserve a solution set and which do not
Two kinds of equation
An equation asserts that two expressions are equal. Whether that assertion is worth investigating depends entirely on how many values make it true, and the source notes open Week 3 by splitting the cases.
True for some values of the variable and false for others. Solving means finding exactly which.
True for every value of the variable. There is nothing to solve; the equation is a statement about the expressions themselves.
| Equation | Kind | Why |
|---|---|---|
| x2 = 9 | Conditional | True at x = 3 and x = -3, false everywhere else |
| (x + d)2 = x2 + 2dx + d2 | Identity | The right side is what the left expands to, so it holds always |
A third case completes the picture: an equation true for no value at all. x + 1 = x + 2 reduces to 1 = 2. There is nothing wrong with the algebra; the assertion is simply false.
Some values
Conditional. Solve it. Solution set is finite or a described infinite set.
All values
Identity. Nothing to solve. Solution set is every admissible value.
No values
Inconsistent. Solution set is empty, ∅.
The solution set
A solution is a value that makes the equation true. The solution set collects all of them, and the source writes it in set notation.
Set notation is worth the small extra effort because it makes the number of solutions explicit. Writing 'x = 3 or x = -3' invites the reader to overlook one; writing {-3, 3} does not.
- {-3, 3}
- Two solutions, listed
- ∅
- No solutions
- ℝ
- Every real number — an identity
- {x : x ≥ 5}
- Infinitely many, described by a condition
An answer is not finished until every solution has been listed. x2 = 9 has two, and writing only x = 3 is a half-answer. The habit of asking 'how many should there be?' before writing down the answer catches most of these.
Equivalent equations
Solving works by replacing an equation with a simpler one that has the same solutions. Two equations with identical solution sets are equivalent, and the whole method rests on knowing which operations preserve equivalence.
| Operation | Preserves solutions? | Note |
|---|---|---|
| Add or subtract the same quantity from both sides | Always | The additive inverse law |
| Multiply both sides by a non-zero constant | Always | The multiplicative inverse law |
| Multiply both sides by an expression in x | Not always | May introduce solutions where the expression is zero |
| Divide both sides by an expression in x | Not always | May lose solutions where the expression is zero |
| Square both sides | Not always | May introduce spurious solutions |
| Take the square root of both sides | Only with ± | Dropping the negative root loses a solution |
From x2 = 3x, dividing both sides by x gives x = 3 and silently discards x = 0, which is a genuine solution. The correct move is to rearrange to x2 - 3x = 0, factor as x(x - 3) = 0, and read off both roots. Never divide by something that might be zero — factor instead.
The source works exactly this trap in Week 6, page 5, on x(2x + 1) = x(x - 4). Cancelling x gives only x = -5; collecting to x(x + 5) = 0 gives x = 0 and x = -5.
Deciding which kind you have
- Simplify both sides as far as possible without moving anything across the equals sign.
- Collect all terms on one side. If everything cancels to 0 = 0, the equation is an identity.
- If a false numeric statement remains — 0 = 3, say — the equation has no solution.
- Otherwise solve the remaining conditional equation normally.
Three quick classifications
| Equation | Collected | Verdict |
|---|---|---|
| 2(x + 3) = 2x + 6 | 0 = 0 | Identity: every real x |
| 2(x + 3) = 2x + 5 | 0 = -1 | No solution: ∅ |
| 2(x + 3) = 3x + 5 | x = 1 | Conditional: {1} |
The three left-hand sides are identical. Only the right sides differ, and each small change moves the equation into a different category.
Common mistakes
| Mistake | Correct | Why |
|---|---|---|
| Reporting one root of x2 = 9 | x = ±3 | Both squares give 9 |
| Dividing by x to solve x2 = 3x | Factor: x(x - 3) = 0 | Division discards x = 0 |
| Treating 0 = 0 as 'no solution' | It is an identity | Everything cancelling means always true |
| Treating 0 = 3 as x = 3 | No solution | The variable has already vanished |
| Squaring without checking afterwards | Always substitute back | Squaring can create solutions that were never there |
Frequently asked questions
Why does the distinction matter?
Because 'solve' means something different in each case. Asking for the solutions of an identity is meaningless — every value works. Asking for the solutions of a conditional equation is the ordinary task.
Can an equation have no solutions at all?
Yes. x + 1 = x + 2 reduces to 1 = 2, which is false whatever x is. The solution set is empty, written ∅. This is the third case alongside 'some values' and 'all values'.
Is (x + d)2 = x2 + 2dx + d2 an equation?
It is an identity. Both sides are the same expression written differently, so it holds for every x. The source notes label it always true.
What makes two equations equivalent?
Having exactly the same solution set. Adding the same quantity to both sides preserves it; squaring both sides may not, which is why radical equations need a check.
Source. Handwritten teaching notes, Week 3, page 1.
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.