Question

Explain how to determine if an equation has infinitely many solutions. Give an example of an ending statement that an equation like this would have.

Answer 1

How to tell if an equation has infinitely many solutions

1. Simplify both sides (expand, combine like terms, move everything to one side).
2. See whether the variable terms cancel out:
- If they cancel and you end up with a true statement (like 0 = 0 or 6 = 6), the original equation is an identity and is true for every allowed value of the variable — so there are infinitely many solutions.
- If they cancel and you end up with a false statement (like 5 = 2), there is no solution.
- If a variable stays, solve normally — you will get one (or a finite number of) solution(s).

Example
Start with 2(x + 3) = 2x + 6.
Expand: 2x + 6 = 2x + 6.
Subtract 2x from both sides: 6 = 6 (true for all x).
Conclusion: the variable cancels leaving a true identity, so the equation has infinitely many solutions.

Example ending statements you could write
- "The equation is an identity; it has infinitely many solutions (all real numbers)."
- "Infinitely many solutions — every real number x satisfies the equation."
- If there are excluded values (e.g., from denominators): "Infinitely many solutions: all real numbers except x = a."