No Solution, or Every Solution

Sometimes solving makes every x disappear. What is left decides the answer.

A false statement

2x + 3 = 2x + 7 becomes 3 = 7. That is false for every x, so there is no solution.

A true statement

2x + 6 = 2(x + 3) becomes 6 = 6. That is true for every x, so every number works.

What it means graphically

No solution means two parallel lines. Every solution means one line drawn twice.

The variable can vanish

When the variable cancels, read what remains. A false statement like 3 = 7 means no solution. A true one like 5 = 5 means every number is a solution. Neither outcome is an error, and both are complete answers.

What it means graphically

Treat each side as a line. No solution means the lines are parallel and never meet; infinitely many means they are the same line. One solution means they cross once. The algebra and the picture always agree.

Identities

An equation true for every value is an identity: 2(x + 3) = 2x + 6. Identities are how algebraic rules are stated. Distinguishing an identity from an equation to be solved is a real conceptual step.

Writing the answer

For no solutions write "no solution" or the empty set symbol; for all values write "all real numbers". Writing x = nothing, or leaving the page blank, does not communicate which case you found.