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Equations with Infinite Solutions and Equations with No Solution
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Not all equations we try to solve will end with $$x$$ = a specific number. Some equations may have infinitely many solutions and other equations may have no solution at all. The following video will show how to recognize these solutions.

Special Cases Equations with Infinite Solutions and Equations with No Solutions

There are three types of answers we can get when solving for a variable:

1. $$x$$ = A specific number (this is what we’ve been getting until now such as $$x = 5.3$$ )
2. $$x$$ = All real numbers or infinitely many solutions (when we get x=x or when any number is equal to itself such as $$3 = 3$$ )
3. No Solutions (when we end with a false statement like $$1 = 5$$ )

### Practice Problems

1. $$-9{\text{M}} {-} 4 = -9{\text{M}} - 4$$ (
Solution
Solution: Infinitely Many Solutions
Details:
In this example, the first thing we need to do is combine like terms. This means we combine the terms with the variable $$M$$ with each other and we combine the terms without a variable together.

Note: There are two versions or ways to solve this equation. Either one is acceptable. You do not have to do both.

First version: Combine terms with variable M first

There is currently a $$−9M$$ on the right-hand side of the equation. We can remove it from the right-hand side and combine it with the left-hand side by adding $$+9M$$ to both sides of the equation.

On the right-hand side:

$$-9M + 9M = 0$$ leaving just $$-4$$

On the left-hand side:

$$-9M + 9M = 0$$ leaving just $$-4$$

Because $$−9M+9M=0$$, we are left with $$−4=−4$$. This statement is always true, therefore, there are infinitely many solutions for the equation $$−9M−4=−9M−4$$. This means that any value of M will still make this equation true

Our final solution: Infinitely many solutions

Second version: Combine terms without a variable first

We want to combine the $$−4$$ on the left-hand side of the equation with the $$−4$$ on the right-hand side of the equation. To do this, add $$+4$$ to both sides of the equation.

On the left-hand side:

$$−4+4=0$$

On the right-hand side:

$$−4+4=0$$

This leaves us with $$−9M=−9M$$. We can either stop here because we see both sides are equal to each other, which means that for any value of M the statement will be true, or we can keep solving for $$M$$.

To keep solving for $$M$$, we need to multiply both sides by the multiplicative inverse of $$−9$$. Multiply both sides by $$-\dfrac{1}{9}$$.

$$\left (-\dfrac{1}{9} \right )\left ( -9 \right )=1$$

This leaves $$1M=1M$$.

$$M=M$$ is always true for any value of $$M$$.

Our final solution: Infinitely Many Solutions
)
2. $$9 + 8{\text{T}} = 13{\text{T}} + 2$$ (
Solution
Solution:
One Solution
)
3. $$-4 + 2{\text{b}} = 2{\text{b}} - 9$$ (
Solution
Solution: No Solution
Details:
We start by combining like terms.

Combine the terms with the variable b by adding $$−2b$$ to both sides of the equation.

Since $$2b+(−2b)=0$$, we are left with $$−4$$ on the left-hand side and $$−9$$ on the right-hand side.

But $$−4$$ does not equal $$−9$$.

This means that no matter what values we put into this equation, it is not true.

Our final solution: No solution
)
4. $$-7 + 7{\text{b}} + 18 = 3{\text{b}} + 3 - 4{\text{b}}$$ (
Solution
Solution:
One Solution
)
5. $$2{\text{x}} + 5 + {\text{x}} = -1 + 3{\text{x}} + 6$$ (
Solution
Solution:
Infinitely Many Solutions
)
6. $$2(3{\text{X}} + 4) = 6{\text{X}} + 7$$ (
Video Solution
Solution: No Solution
Details:

(Video Source | Transcript)
)
7. $$-4(4{\text{M}} {-} 3) = -16{\text{M}} + 12$$ (
Video Solution
Solution: Infinitely Many Solutions
Details:

(Video Source | Transcript)
)

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