Examples · Algebra
Equation Solving Examples: 10 Worked Problems
Ten equations, each adding one difficulty to the last. The final two are the ones that look like mistakes and are not.
One and two steps
The first three equation solving examples need one inverse operation each, then two.
- x + 9 = 49 was added to x
- x = −5subtract 9 from both sides
- 4x = 22x was multiplied by 4
- x = 5.5divide both sides by 4
- 3x − 7 = 14two operations to undo
- 3x = 21add 7 first
- x = 7then divide by 3
Variable on both sides, brackets, fractions
- 3x + 3 = 18subtract 2x from both sides
- 3x = 15subtract 3
- x = 5divide by 3
- 2x + 10 = 3x − 3expand both brackets
- 10 = x − 3subtract 2x
- x = 13add 3
- 3x − 12 = 2xmultiply every term by 12
- x − 12 = 0subtract 2x
- x = 12add 12
Quadratics
- x² = 49both 7 and −7 square to 49
- x = 7 or x = −7take both roots
- x² + 5x + 6 = 0move everything to one side
- (x + 2)(x + 3) = 0factor
- x = −2 or −3set each factor to zero
Example 7 is the one people half-answer. Writing only x = 7 discards a genuine solution, because squaring destroys the sign.
The two that look like errors
Both of these are complete, correct answers.
- 2x + 1 = 2x + 5the equation
- 1 = 5subtract 2x from both sides
- no solutionthe statement is false for every x
- 3x + 6 = 3x + 6expand the bracket
- 0 = 0subtract 3x + 6 from both sides
- every value of xan identity
Nine describes two parallel lines that never meet; ten describes one line written twice. Neither is a failed attempt, and writing 'no solution' or 'all real numbers' is the answer the question wants.
What each example was really testing
The ten worked examples above are ordered by the obstacle each one introduces, not by how hard the arithmetic is.
| Examples | The difficulty | The skill |
|---|---|---|
| 1–3 | one and two inverse operations | isolate the variable |
| 4 | variable on both sides | collect like terms first |
| 5 | brackets | expand before anything else |
| 6 | fractions | multiply every term by the denominator |
| 7–8 | quadratic equation, not linear | factor rather than isolate |
| 9–10 | no solution and identity | read what the cancellation left |
A linear equation yields to isolating the variable; a quadratic equation does not, and recognising which one you are holding is the decision the whole solution rests on.
Checking all ten
Every answer above survives substitution into the original equation.
| Example | Answer | Check |
|---|---|---|
| 3 | x = 7 | 21 − 7 = 14 ✓ |
| 5 | x = 13 | 2(18) = 3(12) → 36 = 36 ✓ |
| 6 | x = 12 | 3 − 1 = 2 ✓ |
| 8 | x = −2 | 4 − 10 = −6 ✓ |
For the quadratics, both roots must be checked. A root that satisfies the rearranged equation but not the original one is an extraneous solution and has to be discarded.
Questions about equation solving examples
How do I solve an equation with fractions?
Multiply every term by the common denominator. The fractions clear in one step, and the equation that remains is an ordinary linear or quadratic one.
Why does x² = 49 have two answers?
Because squaring loses the sign: both 7 and −7 give 49. Taking only the positive root discards half the answer.
What do I write if there is no solution?
Write 'no solution' and show the line where the variable cancelled. That is the complete answer, not an incomplete attempt.
Should I check both roots of a quadratic?
Yes. Rearranging can introduce values that satisfy the new equation but not the original, and only substitution finds them.