Equations With Infinitely Many Solutions

Idea

An equation has infinitely many solutions when both sides are actually the same expression after simplification. That means the equation is true for every value of the variable.

Method

1. Distribute if there are parentheses.
2. Combine like terms on each side.
3. Move variable terms to one side and constants to the other, if needed.
4. Simplify fully.
5. Look at the result:
   • If you get a true statement like 0 = 0, the equation has infinitely many solutions.
   • If you get a false statement like 0 = 5, there is no solution.

How to answer

When the work simplifies to a statement that is always true, the final answer is usually written as all real numbers or infinitely many solutions.

Check

Substitute any value of the variable into the original equation. If it works every time, the equation truly has infinitely many solutions.

Tip

Be careful not to stop too early. Sometimes an equation looks different at first, but after simplifying both sides, all terms cancel. That cancellation is the key sign of infinitely many solutions.