Pre-Algebra
Expressions and equations
Linear equations — one, none, infinite solutions
20 practice questions
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Theory + worked examples
Linear Equations — One, None, or Infinite Solutions
California Pre-Algebra • Standard 8.EE.7 • Expressions & Equations
Linear Equations — One, None, or Infinite Solutions is a topic in Expressions & Equations in the California Common Core State Standards. It is aligned to Standard 8.EE.7, which requires students to solve linear equations and determine the number of solutions.
A linear equation can have one solution, no solution, or infinitely many, depending on what remains after simplifying.
Theory
A linear equation in one variable can have:
- one solution — the variable equals a number,
- no solution — it reduces to a false statement,
- infinitely many — it reduces to a true statement.
If the variables cancel, check whether the leftover statement is true or false.
The three cases.
What the outcome means.
The three outcomes:
\[x=a\ \ (\text{one}),\quad 3=5\ (\text{none}),\quad 4=4\ (\text{infinite})\]
A true numerical statement means all numbers work.
How to classify a linear equation
- Simplify and collect variables on one side.
- If \(x=\) a number, one solution.
- If variables cancel and it is false, no solution.
- If variables cancel and it is true, infinitely many.
Example 1 — One solution
Solve \(2x+1=7\).
Solution
Isolate \(x\).
| \(2x\) | \(=\) | \(6\) |
| \(x\) | \(=\) | \(3\) |
Example 2 — No solution
Solve \(x+2=x+5\).
Solution
Subtract \(x\): \(2=5\) is false.
| \(2\) | \(=\) | \(5\ \text{(false)}\) |
Example 3 — Infinite
Solve \(2(x+1)=2x+2\).
Solution
Both sides equal; \(2=2\) always true.
| \(2x+2\) | \(=\) | \(2x+2\) |
Example 4 — Classify
How many solutions does \(3x=3x-4\) have?
Solution
Subtract \(3x\): \(0=-4\) is false.
Common pitfalls
Cancelling variables does not mean no solution — check the constants.
A true statement means every number is a solution.
A false statement means no solution.
Frequently asked questions
How many solutions can a linear equation have?
One, none, or infinitely many.
What does no solution look like?
A false statement like \(3=5\).
What does infinite solutions look like?
A true statement like \(4=4\).
Solve \(x+2=x+5\).
No solution.
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