Equations
Solving Linear Equations
Isolating the unknown, clearing fractions before anything else, and the transposition shortcut — with every step justified by an inverse operation.
What this page covers
- Solve a linear equation by isolating the variable
- Clear fractions by multiplying through by a common denominator
- Recognise transposition as shorthand for doing the same thing to both sides
- Check a solution by substitution
What counts as linear
The source defines the class by exclusion, which is the clearest way: no x2, no √x, no xy. The variable appears only to the first power, and never inside a radical, a denominator or a product with another variable.
| Equation | Linear? | Reason |
|---|---|---|
| 2x - 3 = 0 | Yes | First power only |
| 2x3 - 7 = x - 36 | Yes | Fractions are fine; the variable is still first power |
| x2 - 9 = 0 | No | Second power |
| √x + 1 = 3 | No | Variable under a radical |
| xy = 6 | No | Product of two variables |
| 1x = 4 | No | Variable in a denominator |
A linear equation in one unknown has exactly one solution, unless the variable cancels entirely — in which case it is either an identity or has no solution.
The method: undo, in reverse order
Solving is unwinding. Whatever was done to the variable is undone by the inverse operation, applied to both sides, working outward from the variable.
Worked example — the simplest case
Each line does exactly one thing to both sides. Writing the intermediate line is worth it while the method is being learned; it makes clear that nothing asymmetric is happening.
2 · 32 - 3 = 3 - 3 = 0. Correct.
Variables on both sides
Collect the variable terms on one side and the constants on the other. Which side takes the variable is a matter of convenience; choosing the side that leaves a positive coefficient saves a sign error later.
Worked example — the source's case
The source collected onto the left, leaving -4x. Collecting onto the right instead gives -17 = 4x and the same answer with one fewer negative to carry.
Left: 3(-174) - 8 = -514 - 324 = -834. Right: 7(-174) + 9 = -1194 + 364 = -834. Equal.
Clearing fractions first
When denominators are present, multiply every term on both sides by a common denominator before doing anything else. The equation becomes integral and the rest is routine.
Worked example — the source's fractional equation
Solve 2x3 - 7 = x - 36.
The denominators are 3 and 6; the lowest common denominator is 6.
The -7 must become -42. Multiplying only the fractional terms is the standard error here and produces a different equation with a different answer.
Left: 263 - 7 = 26 - 213 = 53. Right: 13 - 36 = 106 = 53. Equal.
Transposition
Once the method is secure, the intermediate lines are usually dropped. Moving a term across the equals sign and reversing its sign is called transposing, and the source notes use the word.
| Written | Means | Justified by |
|---|---|---|
| 3x - 8 = 7x becomes 3x = 7x + 8 | Add 8 to both sides | Additive inverse |
| 4x = x + 39 becomes 4x - x = 39 | Subtract x from both sides | Additive inverse |
| 3x = 39 becomes x = 393 | Divide both sides by 3 | Multiplicative inverse |
Transposition is safe for addition and subtraction without qualification. For multiplication and division the factor being moved must be non-zero — which is automatic for a numeric coefficient but is a real condition when the coefficient contains a variable, as it does in Literal Equations.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| Multiplying only the fractional terms | Multiply every term | Count the terms before and after |
| Transposing without reversing the sign | The sign flips | +8 moves across as -8 |
| Distributing a minus over only the first term | -(3y2 - 5y) = -3y2 + 5y | Expand on its own line |
| Dividing by a coefficient that could be zero | State the condition, or factor instead | Ask whether the coefficient can vanish |
| Checking in a partway line rather than the original | Substitute into the original equation | An early error would otherwise be reproduced |
Frequently asked questions
What makes an equation linear?
The variable appears only to the first power. The source notes give the test negatively: no x2, no √x, no xy. A linear equation in one unknown has exactly one solution unless it degenerates.
Should I clear fractions first or last?
First, almost always. Multiplying through by the common denominator turns the whole equation into integers in one step, and the rest is then ordinary arithmetic.
Is transposing the same as doing it to both sides?
Yes — it is an abbreviation, not a separate rule. Moving +8 across as -8 is subtracting 8 from both sides with the intermediate line omitted.
Why check the answer?
It costs seconds and catches sign errors, which are by far the most common failure. Substitute into the original equation, not into a line partway through, or a mistake made early will be reproduced.
Source. Handwritten teaching notes, Week 3, pages 1-2.
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.