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Study Guides > Intermediate Algebra

Read: Classify Solutions to Linear Equations

Learning Objectives

  • 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

Introduction

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 don't 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 examples.

Equations with no solutions

Example

Solve for x. [latex]12+2x–8=7x+5–5x[/latex]

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

[latex] \displaystyle \begin{array}{l}12+2x-8=7x+5-5x\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x+4=2x+5\end{array}[/latex]

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

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

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 [latex]4=5[/latex]. Surely [latex]4[/latex] cannot be equal to [latex]5[/latex]! This may make sense when you consider the second line in the solution where like terms were combined. If you multiply a number by [latex]2[/latex] and add [latex]4[/latex] you would never get the same answer as when you multiply that same number by [latex]2[/latex] and add  [latex]5[/latex]. 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 [latex]x=0[/latex] with “no solution.” The solution [latex]x=0[/latex] means that the value [latex]0[/latex] satisfies the equation, so there is a solution. “No solution” means that there is no value, not even [latex]0[/latex], which would satisfy the equation. Also, be careful not to make the mistake of thinking that the equation [latex]4=5[/latex] means that [latex]4[/latex] and [latex]5[/latex] 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 [latex]12+2x–8=7x+5–5x[/latex] 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 [latex]8y=3(y+4)+y[/latex] 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 [latex]8y=3(y+4)+y[/latex]

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

[latex]8y=3(y+4)+y=8y=3y+12+y[/latex]

Next, begin combining like terms.

[latex]8y=3y+12+y = 8y=4y+12[/latex]

Now move the variable terms to one side. Moving the [latex]4y[/latex] will help avoid a negative sign.

[latex]\begin{array}{l}\,\,\,\,8y=4y+12\\\underline{-4y\,\,-4y}\\\,\,\,\,4y=12\end{array}[/latex]

Now, divide each side by [latex]4y[/latex].

[latex]\begin{array}{c}\Large\frac{4y}{4}\normalsize =\Large\frac{12}{4}\normalsize\\y=3\end{array}[/latex]

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 [latex]2\left(3x-5\right)-4x=2x+7[/latex] 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 [latex]2\left(3x-5\right)-4x=2x+7[/latex]. First, distribute the 2 into the parentheses on the left-hand side.

[latex]\begin{array}{r}2\left(3x-5\right)-4x=2x+7\\6x-10-4x=2x+7\end{array}[/latex]

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

[latex]\begin{array}{r}6x-10-4x=2x+7\\2x-10=2x+7\end{array}[/latex]

Now, take a moment to ponder this equation. It says that [latex]2x-10[/latex] is equal to [latex]2x+7[/latex]. Can some number times two minus 10 be equal to that same number times two plus seven? Let's pretend [latex]x=3[/latex]. Is it true that [latex]2\left(3\right)-10=-4[/latex] is equal to [latex]2\left(3\right)+7=13[/latex]. NO! We don't even really need to continue solving the equation, but we can just to be thorough. Add [latex]10[/latex] to both sides.

[latex]\begin{array}{r}2x-10=2x+7\,\,\\\,\,\underline{+10\,\,\,\,\,\,\,\,\,\,\,+10}\\2x=2x+17\end{array}[/latex]

Now move [latex]2x[/latex] from the right hand side to combine like terms.

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

We know that [latex]0\text{ and }17[/latex] 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 [latex]5x[/latex] and [latex]-4x[/latex] on the left leaves us with an equation with exactly the same terms on both sides of the equal sign.

Example

Solve for x. [latex]5x+3–4x=3+x[/latex]

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

[latex] \displaystyle \begin{array}{r}5x+3-4x=3+x\\x+3=3+x\end{array}[/latex]

Isolate the x term by subtracting x from both sides.

[latex]\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,x+3=3+x\\\,\,\,\,\,\,\,\,\underline{\,-x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-x\,}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\,=\,\,3\end{array}[/latex]

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 “[latex]3=3[/latex].” When you end up with a true statement like this, it means that the solution to the equation is “all real numbers.” Try substituting [latex]x=0[/latex] into the original equation—you will get a true statement! Try [latex]x=-\Large\frac{3}{4}[/latex], and it also will 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 [latex]x+3=3+x[/latex] means “some number plus [latex]3[/latex] is equal to [latex]3[/latex] plus that same number.” We know that this is always true—it’s the commutative property of addition!

Example

Solve for x. [latex]3\left(2x-5\right)=6x-15[/latex]

Answer: Distribute the [latex]3[/latex] through the parentheses on the left-hand side.

[latex] \begin{array}{r}3\left(2x-5\right)=6x-15\\6x-15=6x-15\end{array}[/latex]

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:

[latex]\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,6x-15=6x-15\\\,\,\,\,\,\,\,\,\underline{\,-6x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-6x\,}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-15\,\,=\,\,-15\end{array}[/latex]

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 [latex]0[/latex] 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 x. [latex]7+\left|2x-5\right|=4[/latex]

Answer: Notice absolute value is not alone. Subtract [latex]7[/latex] from each side to isolate the absolute value.

[latex]\begin{array}{r}7+\left|2x-5\right|=4\,\,\,\,\\\underline{\,-7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-7\,}\\\left|2x-5\right|=-3\end{array}[/latex]

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 x. [latex]-\Large\frac{1}{2}\normalsize\left|x+3\right|=6[/latex]

Answer: Notice absolute value is not alone, multiply both sides by the reciprocal of [latex]-\Large\frac{1}{2}[/latex], which is [latex]-2[/latex].

[latex]\begin{array}{r}-\Large\frac{1}{2}\normalsize\left|x+3\right|=6\,\,\,\,\,\,\,\,\,\,\,\,\\\,\,\,\,\,\,\,\,\left(-2\right)-\Large\frac{1}{2}\normalsize\left|x+3\right|=\left(-2\right)6\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left|x+3\right|=-12\,\,\,\,\,\end{array}[/latex]

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
And sometimes, we don't need to do much algebra to see what the outcome will be.

Licenses & Attributions

CC licensed content, Original

  • Revision and Adaptation. Provided by: Lumen Learning License: Public Domain: No Known Copyright.
  • Absolute Value Equations with No Solutions. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Linear Equations with No Solutions or Infinite Solutions. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Linear Equations with No Solutions of Infinite Solutions (Parentheses). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Absolute Value Equations with No Solutions. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.

CC licensed content, Shared previously

  • Beginning and Intermediate Algebra. Authored by: Tyler Wallace. Located at: http://www.wallace.ccfaculty.org/book/book.html. License: CC BY: Attribution.
  • Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.