# 2.5.1: Applications of Linear Equations

- Page ID
- 4368

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

Suppose you have 115 connections on a social networking website, which is 28 more than your friend has. How many connections does your friend have? How about if you had 5 times as many connections as your friend?

**Applications of One-Step Equations**

Many careers base their work on manipulating linear equations. Consider the botanist studying bamboo as a renewable resource. She knows bamboo can grow up to 60 centimeters per day. If the specimen she measured was 1 meter tall, how long would it take to reach 5 meters in height? By writing and solving this equation, she will know exactly how long it should take for the bamboo to reach the desired height.

**Let's solve the following problems using one step equations:**

- One method to weigh a horse is to load it into an empty trailer with a known weight and reweigh the trailer. A Shetland pony is loaded onto a trailer that weighs 2,200 pounds empty. The trailer is then reweighed. The new weight is 2,550 pounds. How much does the pony weigh?

Choose a variable to represent the weight of the pony, say p.

Write an equation: 2550=2200+p.

Apply the **Addition Property of Equality**: 2550−2200=2200+p−2200.

Simplify: 350=p.

The Shetland pony weighs 350 pounds.

- In good weather, tomato seeds can grow into plants and bear ripe fruit in as few as 19 weeks. Lorna planted her seeds 11 weeks ago. How long must she wait before her tomatoes are ready to be picked?

The variable in question is the number of weeks until the tomatoes are ready. Call this variable w.

Write an equation: w+11=19.

Solve for w by using the Addition Property of Equality.

w+11−11=19−11

w=8

It will take as few as 8 weeks for the plant to bear ripe fruit.

- In 2004, Takeru Kobayashi of Nagano, Japan, ate 53(1/2) hot dogs in 1/2 minutes. He broke his previous world record, set in 2002, by three hot dogs. Calculate:
- How many minutes it took him to eat one hot dog.
- How many hot dogs he ate per minute.
- What his old record was.

Write an equation, letting m represent the number of minutes to eat one hot dog: 53.5m=12.

Applying the **Multiplication Property of Equality**:

53.5m/53.5=12/53.5

m=0.224 minutes

It took approximately 0.224 minutes, or 13.44 seconds, to eat one hot dog.

Questions (b) and (c) are left for you to complete in the Review (#3).

### Examples

Example 2.5.1.1

Earlier, you were told that you have 115 connections on a social networking website, which is 28 more than your friend has. How many connections does your friend have? How about if you had 5 times as many connections as your friend?

**Solution**

Let's use c to represent the number of connections your friend has. If you have 28 more connections than your friend, then the equation that represents the situation is:

115=28+c

Now, use the Addition Property of Equality:

115=28+c

115−28=28+c−28

87=c

You friend has 87 connections.

If you have 3 times as many connections as your friend, then the equation that represents the situation is:

115=3c

Now, use the Multiplication Property of Equality:

115=5c

115÷5=5c÷5

23=c

Your friend has 23 connections.

Example 2.5.1.2

Mayra can run 6.5 miles per hour. If Mayra runs for 2-and-a-quarter hours, how far will she have gone?

**Solution**

We can use the formula for speed: speed=distancetime.

Substituting in speed=6.5 and time=2.25 we get:

6.5=distance3.25.

Now, use the Multiplication Property of Equality:

6.5×2.25=distance2.25×2.256.5×2.25=distance2.25×2.256.5×2.25=distance13.5=distance

Mayra can run 13.5 miles in 2-and-a-quarter hours.

### Review

- Peter is collecting tokens on breakfast cereal packets in order to get a model boat. In eight weeks he has collected 10 tokens. He needs 25 tokens for the boat. Write an equation and determine the following information.
- How many more tokens he needs to collect, n.
- How many tokens he collects per week, w.
- How many more weeks remain until he can send off for his boat, r.

- Juan has baked a cake and wants to sell it in his bakery. He is going to cut it into 12 slices and sell them individually. He wants to sell it for three times the cost of making it. The ingredients cost him $8.50, and he allowed $1.25 to cover the cost of electricity to bake it. Write equations that describe the following statements.
- The amount of money that he sells the cake for (u).
- The amount of money he charges for each slice (c).
- The total profit he makes on the cake (w).

- Solve the remaining two questions regarding Takeru Kobayashi in Problem 3 from the Applications of One-Step Equations section.

**Mixed Review**

- Simplify √48.
- Classify 6.23 according to the real number chart.
- Reduce 1184.
- Graph the following ordered pairs: {(2,−2),(4,−1),(5,−5),(3,−2)}.
- Define evaluate.
- Underline the math verb in this sentence: m minus n is 16.
- What property is illustrated here? 4(a+11.2)=4(a)+4(11.2)

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.2.

### Vocabulary

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

Addition Property of Equality |
For all real numbers a,b, and c: If a=b, then a+c=b+c. |

Multiplication Property of Equality |
For all real numbers a,b, and c: a=b, If a(c)=b(c).then |

### Additional Resources

Activity: **Applications of One-Step Equations Discussion Questions**

Practice: **Applications of Linear Equations**

Real World Application: **The Cost of College**