We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Study Guides > Intermediate Algebra

Classify Solutions to Linear Equations

Learning Outcomes

  • Solve equations that have one solution, no solution, or an infinite number of solutions
  • Recognize when a linear equation that contains absolute value does not have a solution
There are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that do not have any solutions and even some that have an infinite number of solutions. The case where an equation has no solution is illustrated in the next example.

Equations with No Solutions

Example

Solve for x12+2x8=7x+55x12+2x–8=7x+5–5x

Answer: Combine like terms on both sides of the equation.

12+2x8=7x+55x              2x+4=2x+5 \displaystyle \begin{array}{l}12+2x-8=7x+5-5x\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x+4=2x+5\end{array}

Isolate the x term by subtracting 2x from both sides.

            2x+4=2x+5        2x          2x                                4=5\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,2x+4=2x+5\\\,\,\,\,\,\,\,\,\underline{-2x\,\,\,\,\,\,\,\,\,\,-2x\,\,\,\,\,\,\,\,}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4= \,5\end{array}

This false statement implies there are no solutions to this equation. Sometimes, we say the solution does not exist, or DNE for short.

This is not a solution! You did not find a value for x. Solving for x the way you know how, you arrive at the false statement 4=54=5. Surely 44 cannot be equal to 55! This may make sense when you consider the second line in the solution where like terms were combined. If you multiply a number by 22 and add 44 you would never get the same answer as when you multiply that same number by 22 and add  55. Since there is no value of x that will ever make this a true statement, the solution to the equation above is “no solution.” Be careful that you do not confuse the solution x=0x=0 with “no solution.” The solution x=0x=0 means that the value 00 satisfies the equation, so there is a solution. “No solution” means that there is no value, not even 00, which would satisfy the equation. Also, be careful not to make the mistake of thinking that the equation 4=54=5 means that 44 and 55 are values for x that are solutions. If you substitute these values into the original equation, you’ll see that they do not satisfy the equation. This is because there is truly no solution—there are no values for x that will make the equation 12+2x8=7x+55x12+2x–8=7x+5–5x true.

Think About It

Try solving these equations. How many steps do you need to take before you can tell whether the equation has no solution or one solution? a) Solve 8y=3(y+4)+y8y=3(y+4)+y Use the textbox below to record how many steps you think it will take before you can tell whether there is no solution or one solution. [practice-area rows="1"][/practice-area]

Answer:

Solve 8y=3(y+4)+y8y=3(y+4)+y

First, distribute the 3 into the parentheses on the right-hand side.

8y=3y+12+y8y=3y+12+y

Next, begin combining like terms.

8y=4y+128y=4y+12

Now move the variable terms to one side. Moving the 4y4y will help avoid a negative sign.

    8y=4y+124y  4y    4y=12\begin{array}{l}\,\,\,\,8y=4y+12\\\underline{-4y\,\,-4y}\\\,\,\,\,4y=12\end{array}

Now, divide each side by 4y4y.

4y4=124y=3\begin{array}{c}\dfrac{4y}{4}\normalsize =\dfrac{12}{4}\normalsize\\y=3\end{array}

Because we were able to isolate y on one side and a number on the other side, we have one solution to this equation.

b) Solve 2(3x5)4x=2x+72\left(3x-5\right)-4x=2x+7 Use the textbox below to record how many steps you think it will take before you can tell whether there is no solution or one solution. [practice-area rows="1"][/practice-area]

Answer: Solve 2(3x5)4x=2x+72\left(3x-5\right)-4x=2x+7. First, distribute the 2 into the parentheses on the left-hand side.

6x104x=2x+7\begin{array}{r}6x-10-4x=2x+7\end{array}

Now begin simplifying. You can combine the x terms on the left-hand side.

2x10=2x+7\begin{array}{r}2x-10=2x+7\end{array}

Now, take a moment to ponder this equation. It says that 2x102x-10 is equal to 2x+72x+7. Can some number times two minus 10 be equal to that same number times two plus seven? Pretend x=3x=3. Is it true that 2(3)10=42\left(3\right)-10=-4 is equal to 2(3)+7=132\left(3\right)+7=13. NO! We do not even really need to continue solving the equation, but we can just to be thorough. Add 1010 to both sides.

2x10=2x+7    +10           +102x=2x+17\begin{array}{r}2x-10=2x+7\,\,\\\,\,\underline{+10\,\,\,\,\,\,\,\,\,\,\,+10}\\2x=2x+17\end{array}

Now subtract 2x2x from both sides.

     2x=2x+17  2x  2x       0=17\begin{array}{l}\,\,\,\,\,2x=2x+17\\\,\,\underline{-2x\,\,-2x}\\\,\,\,\,\,\,\,0=17\end{array}

We know that 0 and 170\text{ and }17 are not equal, so there is no number that x could be to make this equation true. This false statement implies there are no solutions to this equation, or DNE (does not exist) for short.

Algebraic Equations with an Infinite Number of Solutions

You have seen that if an equation has no solution, you end up with a false statement instead of a value for x. It is possible to have an equation where any value for x will provide a solution to the equation. In the example below, notice how combining the terms 5x5x and 4x-4x on the left leaves us with an equation with exactly the same terms on both sides of the equal sign.

Example

Solve for x5x+34x=3+x5x+3–4x=3+x

Answer: Combine like terms on both sides of the equation.

5x+34x=3+xx+3=3+x \displaystyle \begin{array}{r}5x+3-4x=3+x\\x+3=3+x\end{array}

Isolate the x term by subtracting x from both sides.

             x+3=3+x        x                    x                     3  =  3\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,x+3=3+x\\\,\,\,\,\,\,\,\,\underline{\,-x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-x\,}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\,=\,\,3\end{array}

This true statement implies there are an infinite number of solutions to this equation, or we can also write the solution as "All Real Numbers"

You arrive at the true statement “3=33=3.” When you end up with a true statement like this, it means that the solution to the equation is “all real numbers.” Try substituting x=0x=0 into the original equation—you will get a true statement! Try x=34x=-\dfrac{3}{4} and it will also check! This equation happens to have an infinite number of solutions. Any value for x that you can think of will make this equation true. When you think about the context of the problem, this makes sense—the equation x+3=3+xx+3=3+x means “some number plus 33 is equal to 33 plus that same number.” We know that this is always true—it is the commutative property of addition!

Example

Solve for x3(2x5)=6x153\left(2x-5\right)=6x-15

Answer: Distribute the 33 through the parentheses on the left-hand side.

3(2x5)=6x156x15=6x15 \begin{array}{r}3\left(2x-5\right)=6x-15\\6x-15=6x-15\end{array}

Wait! This looks just like the previous example. You have the same expression on both sides of an equal sign.  No matter what number you choose for x, you will have a true statement. We can finish the algebra:

             6x15=6x15        6x               6x                   15  =  15\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,6x-15=6x-15\\\,\,\,\,\,\,\,\,\underline{\,-6x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-6x\,}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-15\,\,=\,\,-15\end{array}

This true statement implies there are an infinite number of solutions to this equation.

In the following video, we show more examples of attempting to solve a linear equation with either no solution or many solutions. https://youtu.be/iLkZ3o4wVxU In the following video, we show more examples of solving linear equations with parentheses that have either no solution or many solutions. https://youtu.be/EU_NEo1QBJ0

Absolute Value Equations with No Solutions

As we are solving absolute value equations, it is important to be aware of special cases. An absolute value is defined as the distance from 00 on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples.

Example

Solve for x7+2x5=47+\left|2x-5\right|=4

Answer: Notice absolute value is not alone. Subtract 77 from each side to isolate the absolute value.

7+2x5=4    7                          72x5=3\begin{array}{r}7+\left|2x-5\right|=4\,\,\,\,\\\underline{\,-7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-7\,}\\\left|2x-5\right|=-3\end{array}

Result of absolute value is negative! The result of an absolute value must always be positive, so we say there is no solution to this equation, or DNE.

Example

Solve for x12x+3=6-\dfrac{1}{2}\normalsize\left|x+3\right|=6

Answer: Notice absolute value is not alone. Multiply both sides by the reciprocal of 12-\dfrac{1}{2}, which is 2-2.

12x+3=6                    (2)12x+3=(2)6                     x+3=12     \begin{array}{r}-\dfrac{1}{2}\normalsize\left|x+3\right|=6\,\,\,\,\,\,\,\,\,\,\,\,\\\,\,\,\,\,\,\,\,\left(-2\right)-\dfrac{1}{2}\normalsize\left|x+3\right|=\left(-2\right)6\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left|x+3\right|=-12\,\,\,\,\,\end{array}

Again, we have a result where an absolute value is negative! There is no solution to this equation, or DNE.

In this last video, we show more examples of absolute value equations that have no solutions. https://youtu.be/T-z5cQ58I_g

Summary

We have seen that solutions to equations can fall into three categories:
  • One solution
  • No solution, DNE (does not exist)
  • Many solutions, also called infinitely many solutions or All Real Numbers
Keep in mind that sometimes we do not need to do much algebra to see what the outcome will be.

Licenses & Attributions