198
5.4 Solving Equations with Infinite or No Solutions
So far we have looked at equations where there is exactly one solution. It is possible to have more than
solution in other types of equations that are not linear, but it is also possible to have no solutions or infinite
solutions. No solution would mean that there is no answer to the equation. It is impossible for the equation to be
true no matter what value we assign to the variable. Infinite solutions would mean that any value for the variable
would make the equation true.
No Solution Equations
Let’s look at the following equation:
2 3 2 7
Note that we have variables on both sides of the equation. So we’ll subtract 2 from both sides to
eliminate the 2 on the right side of the equation. However, something odd happens.
That can’t be right! We know that three doesn’t equal seven. It is a false statement to say 3 7, so we
know that there can be no solution. Does that make sense though? Well if we took twice a number and added
three, would it ever be the same as twice a number and adding seven?
Let’s look at another example equation:
3 4 3 11
Note that we need to simplify and that there are variables on both sides of the equation. So we’ll first
multiply through the parentheses with the distributive property and then subtract 3 from both sides to eliminate
the 3 on the right side of the equation.
We again get a false statement and therefore we know there are no solutions. Sometimes we use the
symbol Ø to represent no solutions. That symbol means “empty set” which means that the set of all answers is
empty. In other words, there is no answer. So if we want to use Ø to represent no solution, we may.
2
3
2
7
2
2
3
7
3
3
12
11
3
4
3
11
3
12
3
11
