# 2.2.6: Checking Solutions to Equations

- Page ID
- 4356

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Evaluating Algebraic Expressions and Equations

Your summer landscaping job pays a fixed rate of $20 per job plus $4 an hour. How much total money would you would make if it takes you 3 hours to complete a single job?

### Evaluating Algebraic Expressions and Equations

You have probably seen letters in a mathematical **expression**, such as 3x−8. These letters, also called **variables,** represent an unknown number. One of the goals of algebra is to solve various equations for a **variable**. Typically, x is used to represent the unknown number, but any letter can be used.

To evaluate an expression or **equation**, we would need to substitute in a given value for the variable and test it. In order for the given value to be true for an equation, the two sides of the equation must simplify to the same number.

Let's evaluate the following expressions and equations.

- Evaluate 2x
^{2}−9 for x=−3.

We know that 2x^{2}−9 is an expression because it does not have an equals sign. Therefore, to evaluate this expression, plug in -3 for x and simplify using the Order of Operations.

2(−3)^{2}−9→(−3)^{2}=−3⋅−3=9

2(9)−9

18−9

9

You will need to remember that when squaring a negative number, the answer will always be positive. There are three different ways to write multiplication: 2×9,2⋅9, and 2(9).

- Determine if x=5 is a
**solution**to 3x−11=14.

Even though the directions are different, this problem is almost identical to #1 above. However, this is an equation because of the equals sign. Both sides of an equation must be equal to each other in order for it to be true. Plug in 5 everywhere there is an x. Then, determine if both sides are the same.

3(5)−11=14

15−11≠14

4≠14

Because 4≠14, this is not a true equation. Therefore, 5 is not a solution.

- Determine if t=−2 is a solution to 7t
^{2}−9t−10=36.

Here, t is the variable and it is listed twice in this equation. Plug in -2 everywhere there is a tand simplify.

7(−2)^{2}−9(−2)−10=36

7(4)+18−10=36

28+18−10=36

36=36

-2 is a solution to this equation.

**Examples**

Example \(\PageIndex{1}\)

Earlier, you were given a problem about your summer landscaping job.

Rewrite the sentence as an algebraic expression. $20 plus $4 an hour, would be 20+4h, where * h* equals the number of hours you work. Then, evaluate the expression for h=3.

**Solution**

20+4(3)=20+12=32

You will make a total of $32 for this particular job.

Example \(\PageIndex{1}\)

Evaluate s^{3}−5s+6 for s=4

**Solution**

Plug in 4 everywhere there is an s.

4^{3}−5(4)+6

64−20+6

50

Example \(\PageIndex{1}\)

Determine if a=−1 is a solution to 4a−a^{2}+11=−2−2a.

**Solution**

Plug in -1 for a and see if both sides of the equation are the same.

4(−1)−(−1)^{2}+11=−2−2(−1)

−4−1+11=−2+2

6≠0

Because the two sides are not equal, -1 is not a solution to this equation.

**Review**

Evaluate the following expressions for x=5.

- 4x−11
- x
^{2}+8 - 1/2x+1

Evaluate the following expressions for the given value.

- −2a+7;a=−1
- 3t
^{2}−4t+5;t=4 - (2/3)c−7;c=−9
- x
^{2}−5x+6;x=3 - 8p
^{2}−3p−15;p=−2 - m
^{3}−1;m=1

Determine if the given values are solutions to the equations below.

- x
^{2}−5x+4=0;x=4 - y
^{3}−7=y+3;x=2 - 7x−3=4;x=1
- 6z+z−5=2z+12;z=−3
- 2b−5b
^{2}+1=b^{2};b=6 - −(1/4)g+9=g+15;g=−8

Find the value of each expression, given that a=−1,b=2,c=−4, and d=0.

- ab−c
- b
^{2}+2d - c+(1/2)b−a
- b(a+c)−d
^{2}

For problems 20-25, use the equation y^{2}+y−12=0.

- Is y=4 a solution to this equation?
- Is y=−4 a solution to this equation?
- Is y=3 a solution to this equation?
- Is y=−3 a solution to this equation?
- Do you think there are any other solutions to this equation, other than the ones found above?
**Challenge**Using the solutions you found from problems 20-23, find the sum of these solutions and their product. What do you notice?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.4

### Vocabulary

Term | Definition |
---|---|

Equation |
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs. |

Expression |
An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols. |

inequality |
An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are <, >, ≤, ≥ and ≠. |

solution |
A solution to an equation or inequality should result in a true statement when substituted for the variable in the equation or inequality. |

Variable |
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n. |

### Additional Resources

Video:

PLIX: Play, Learn, Interact, eXplore: **Checking Solutions to Equations: Taxi Cab Calculations**

Real World Application: **Is It a Bear or a Bull?**

Practice: **Checking Solutions to Equations**