Equations With No Solution

What “no solution” means

An equation has no solution when simplifying it leads to a statement that is false, such as 0 = 5. That means there is no value of the variable that makes the equation true.

How to solve

1. Simplify both sides: distribute, combine like terms, and remove parentheses if needed.
2. Get all variable terms on one side and constants on the other.
3. Combine like terms carefully.
4. Watch for the key result:
   • If the variable disappears and you get a false statement, the equation has no solution.
   • Example of a false statement: 3 = 7.

Why this happens

Sometimes the variable terms cancel out, leaving only numbers. If those numbers are different, the equation cannot be true for any value of the variable.

Check your work

Substitute any number only to see whether the original equation can ever work, but the final decision comes from simplification. A correct no-solution result is usually written as no solution or ∅.

Common mistake

Do not stop too early. An equation may look solvable before simplification, but after combining terms it can become a contradiction.