A linear equation has no solution when simplifying both sides eliminates the variable but leaves a false statement, such as , because no value of can make the constants equal. This corresponds to distinct parallel lines in a graph and contrasts with equations yielding one solution or an identity with infinitely many solutions; more advanced systems and nonlinear cases are not included.
To identify an equation with no solution, simplify it until the variable is eliminated.
Example:
Subtract from both sides:
The variable has disappeared, but the statement is false. Therefore, the equation has no solution.
The two sides have the same -term but different constants, so they can never be equal.
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