Method · Algebra
How to Solve Equations: Step by Step
One procedure handles almost every equation you will meet. The order of the steps is what makes it work.
The procedure, in order
Knowing how to solve equations is mostly knowing this order and not skipping ahead. Each step removes one kind of obstacle, and later steps assume the earlier ones are done.
- Clear any fractions by multiplying every term by the common denominator.
- Expand any brackets.
- Collect like terms on each side separately.
- Move the variable terms to one side, the numbers to the other.
- Divide by the coefficient of the variable.
- Check by substituting back into the original.
A straightforward one
- 5x + 3 = 2x + 18variable on both sides
- 3x + 3 = 18subtract 2x from both sides
- 3x = 15subtract 3 from both sides
- x = 5divide both sides by 3
Moving the smaller x term is usually easier, because it leaves a positive coefficient. Subtracting 5x instead would give −3x = −15, which is correct but invites a sign error at the last step.
With brackets
Expand first. Trying to divide before the bracket is gone is where this goes wrong.
- 3(x − 4) = 2x + 1the equation
- 3x − 12 = 2x + 1expand: multiply BOTH terms in the bracket
- x − 12 = 1subtract 2x from both sides
- x = 13add 12 to both sides
The expansion is where marks are lost. 3(x − 4) is 3x − 12, not 3x − 4: the multiplier reaches every term inside the bracket.
With fractions
Multiply every term by the common denominator and the fractions disappear in one move.
- x⁄2 + x⁄3 = 5denominators 2 and 3
- 3x + 2x = 30multiply EVERY term by 6
- 5x = 30collect like terms
- x = 6divide both sides by 5
The 5 on the right had to be multiplied by 6 as well. Multiplying only the terms that contain fractions is the classic slip here, and it changes the equation.
When it turns out to be quadratic
If an x² survives, stop trying to isolate x. Move everything to one side and factor or use the formula.
- x² + 5x = −6an x² term, so this is quadratic
- x² + 5x + 6 = 0move every term to the left
- (x + 2)(x + 3) = 0factor
- x = −2 or x = −3a product is zero when a factor is
Two roots is normal here, not a sign that something went wrong. Squaring loses the sign, so a quadratic usually has two answers.
The inverse operations, listed
Every step in a linear solution is one inverse operation applied to both sides. There are only four to know.
| Built by | Undone by |
|---|---|
| addition | subtraction |
| subtraction | addition |
| multiplication | division |
| division | multiplication |
To isolate the variable you apply them in reverse order to how the expression was built. In 3x − 7, x was multiplied then had 7 subtracted, so you add 7 first and divide by 3 second. Reversing that order is the most common structural error in an otherwise correct solution.
If the variable cancels entirely along the way, stop and read what is left. A true statement means every value works; a false one means no solution.
Check every answer
Substitute back into the original equation — the one as it was written, before any rearranging.
| Equation | Answer | Check |
|---|---|---|
| 5x + 3 = 2x + 18 | x = 5 | 28 = 28 ✓ |
| 3(x − 4) = 2x + 1 | x = 13 | 27 = 27 ✓ |
| x/2 + x/3 = 5 | x = 6 | 3 + 2 = 5 ✓ |
Checking against the original rather than an intermediate line is deliberate. If step two contained an error, every line after it is internally consistent and still wrong.
Questions about how to solve equations
What is the first step in solving an equation?
Clear fractions and expand brackets. Only once the equation is free of both should you start collecting terms and moving things across.
What do I do when the variable is on both sides?
Subtract the smaller variable term from both sides so it appears on one side only. Choosing the smaller one keeps the coefficient positive.
How do I know if an equation is linear or quadratic?
Look for an x² after expanding and collecting. If one survives, the equation is quadratic and needs factoring or the quadratic formula.
Why check against the original equation?
Because an error made early is carried forward consistently. Checking a later line only confirms the arithmetic after the mistake, not before it.