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学習ガイド > College Algebra CoRequisite Course

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.

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.

In the example above, a solution was not obtained. Using the the properties of equality to isolate the variable resulted instead in the false statement [latex]4=5[/latex]. Certainly, [latex]4[/latex] is not equal to [latex]5[/latex]. Note that in the second line of the solution above, the statement [latex]2x+4=2x+5[/latex] was obtained after combining like terms on both sides. If we examine that statement carefully, we can see that it was false even before we attempted to solve it. It would not be possible for the quantity [latex]2x[/latex] with [latex]4[/latex] added to it to be equal to the same quantity [latex]2x[/latex] with [latex]5[/latex] added to it. The two sides of the equation do not balance. Since there is no value of x that will ever make this a true statement, we say that the equation has 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. To say that a statement has no solution means that there is no value of the variable, not even [latex]0[/latex], which would satisfy the equation (that is, make the original statement 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=3y+12+y[/latex]

Next, begin combining like terms.

[latex]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}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}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? 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 do not 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 subtract [latex]2x[/latex] from both sides.

[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.

Equations with Many 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"

When solving, the true statement “[latex]3=3[/latex]” was obtained. When solving an equation reveals a true statement like this, it means that the solution to the equation is all real numbers, that is, there are infinitely many solutions. Try substituting [latex]x=0[/latex] into the original equation—you will get a true statement! Try [latex]x=-\dfrac{3}{4}[/latex]. It will also satisfy the equation. In fact any real value of x will make the original statement true. Indeed, after combining like terms, the equation [latex]x+3=3+x[/latex] was obtained. It is certainly true that the quantity [latex]x[/latex] with [latex]3[/latex] added to it is equal to [latex]3[/latex] with [latex]x[/latex] added to it by 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.

Watch the following video for demonstrations of equations with no solutions and infinitely many solutions. https://youtu.be/iLkZ3o4wVxU The next video demonstrates equations with no or infinitely many solutions involving parentheses. 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  of a number from [latex]0[/latex] on a number line, so the absolute value of a number must be a positive. When an absolute value expression is given to be equal to a negative number, we say the equation has no solution (DNE, for short). 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, see show more examples of absolute value equations that have no solutions. https://youtu.be/T-z5cQ58I_g We have seen that solutions to equations can fall into three categories:
  1. exactly one solution;
  2. no solution  (also called DNE for does not exist)); or
  3. many solutions (also called infinitely many solutions, or we may say the solution is 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

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.