Solving for a Variable:

Equations with Infinite Solutions and Equations with No Solution

Not all equations we try to solve will end with ${\text{x}}$ = 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.

Video Source (05:35 mins) | Transcript

There are 3 types of answers we can get when solving for a variable:

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

Additional Resources

Khan Academy: Number of Solutions to Equations (05:26 mins, Transcript)
Khan Academy: Number of Solutions - Example (01:54 mins, Transcript)

Practice Problems

$-9{\text{M}} {-} 4 = -9{\text{M}} - 4$

$9 + 8{\text{T}} = 13{\text{T}} + 2$

$-4 + 2{\text{b}} = 2{\text{b}} - 9$

$-7 + 7{\text{b}} + 18 = 3{\text{b}} + 3 - 4{\text{b}}$

$2{\text{x}} + 5 + {\text{x}} = -1 + 3{\text{x}} + 6$

$2(3{\text{X}} + 4) = 6{\text{X}} + 7$

$-4(4{\text{M}} {-} 3) = -16{\text{M}} + 12$

Solutions

Infinitely Many Solutions (Written Solution)

In this example, the first thing we need to do is combine like terms. This means we combine the terms with the variable ${\text{M}}$ 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 ${\text{M}}$ first

There is currently a $-9{\text{M}}$ 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 $+9{\text{M}}$ to both sides of the equation

On the right-hand side:

$-9{\text{M}} + 9{\text{M}} = 0$ leaving just $-4$

On the left-hand side:

$-9{\text{M}} + 9{\text{M}} = 0$ leaving just $-4$

Because $-9{\text{M}} + 9{\text{M}} = 0$, we are left with $-4 = -4$. This statement is always true, therefore, there are infinitely many solutions for the equation $-9{\text{M}} {-} 4 = -9{\text{M}} - 4$. This means that any value of ${\text{M}}$ 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 $-9{\text{M}}=-9{\text{M}}$. We can either stop here because we see both sides are equal to each other, which means that for any value of ${\text{M}}$ the statement will be true, or we can keep solving for ${\text{M}}$.

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

$This image is (negative one-ninth) multiplied to (negative nine M) equals (negative one-ninth) multiplied to (negative nine M). The fractions one-ninth on both sides of the equal sign are green, indicating they are new.$

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

This leaves $1{\text{M}} = 1{\text{M}}$.

$This image has three equations listed one above the other. The first equation is (negative one-ninth) multiplied to (negative nine M) equals (negative one-ninth) multiplied to (negative nine M). The fractions one-ninth on both sides of the equal sign are green. The equation below this is the same except the negative 9 is separated by parentheses from the M. (Negative one-ninth) multiplied to (negative nine) multiplied to M equals (negative one-ninth) multiplied to (negative nine) multiplied to M. Below the (negative one-ninth) multiplied to (negative nine) there is a blue bracket grouping them together and the point of the blue bracket points to blue text that says, “Multiplicative inverses =1.” This is the same on both sides of the equal sign. Below all of this s the last equation which is a blue number 1 multiplied to M equals a blue number 1 multiplied to M.$

${\text{M}}={\text{M}}$ is always true for any value of ${\text{M}}$.

Our final solution: Infinitely Many Solutions

One Solution

No Solution (Written Solution)

We start by combining like terms.

Combine the terms with the variable b by adding $-2{\text{b}}$ to both sides of the equation.

Since $2{\text{b}} + (-2{\text{b}}) = 0$, 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

One Solution

Infinitely Many Solutions

No Solution (Solution Video | Transcript)

Infinitely Many Solutions (Solution Video | Transcript)