Algebra
Single-variable equations
Equations with no/infinite solutions
20 practice questions
2 video lessons
Theory + worked examples
Theory
When solving a linear equation, the variables may cancel, giving one of three outcomes:
- One solution: you get \(x=\) a number.
- No solution: a false statement like \(3=5\).
- Infinitely many: an always-true statement like \(0=0\).
No solution means the lines are parallel; infinite means they are the same line.
The three possible outcomes.
Parallel lines give no solution.
Interpreting the result:
\[a=a\ \Rightarrow\ \text{infinite},\qquad a=b\ (a\neq b)\ \Rightarrow\ \text{none}\]
If the variable disappears, read the truth of what remains.
How to classify
- Simplify both sides.
- Collect variable terms.
- If the variable cancels, read the statement.
- True \(\Rightarrow\) infinite; false \(\Rightarrow\) none.
Example 1 — No solution
Solve \(2x+3=2x+5\).
Solution
Subtract \(2x\) from both sides.
| \(3\) | \(=\) | \(5\ \text{(false)}\) |
No value works — no solution.
Example 2 — Infinitely many
Solve \(3(x+2)=3x+6\).
Solution
Distribute and simplify.
| \(3x+6\) | \(=\) | \(3x+6\) |
| \(0\) | \(=\) | \(0\ \text{(always true)}\) |
Every value works — infinitely many.
Example 3 — One solution
Solve \(4x-1=2x+7\).
Solution
Collect variables.
| \(2x\) | \(=\) | \(8\) |
| \(x\) | \(=\) | \(4\) |
Exactly one solution.
Example 4 — Recognize the type
How do you know an equation has no solution?
Solution
The variables cancel and you get a false statement like \(3=5\).
Common pitfalls
If the variable cancels, check whether what remains is true.
\(0=0\) is infinite, not “no solution.”
\(3=5\) is no solution, not \(x=0\).
Frequently asked questions
When does an equation have no solution?
When the variables cancel and a false statement remains.
When does it have infinitely many solutions?
When both sides are identical, giving an always-true statement.
What does \(0=0\) mean?
Infinitely many solutions.
What does \(2=7\) mean?
No solution.
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