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# 3.2: Applications of One-Step Equations

Difficulty Level: Basic Created by: CK-12
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Practice Applications of One-Step Equations

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Suppose you have 115 connections on a social networking website, which is 28 more than your friend has. If you represented the number of your friend's connections with the variable c\begin{align*}c\end{align*}, could you write an equation to represent this situation and solve it in only one step? How about if you had 3 times as many connections as your friend? After completing this Concept, you'll be able to write equations for both of these scenarios and solve each of them in a single step.

### Guidance

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. In this Concept, you will see examples where one-step linear equations are used to solve real-world problems.

#### Example A

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?

Solution: Choose a variable to represent the weight of the pony, say p\begin{align*}p\end{align*}.

Write an equation: 2550=2200+p\begin{align*}2550 = 2200 + p\end{align*}.

Apply the Addition Property of Equality: 25502200=2200+p2200.\begin{align*}2550 - 2200 = 2200 + p - 2200.\end{align*}

Simplify: 350=p\begin{align*}350 = p\end{align*}.

The Shetland pony weighs 350 pounds.

#### Example B

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?

Solution: The variable in question is the number of weeks until the tomatoes are ready. Call this variable w\begin{align*}w\end{align*}.

Write an equation: w+11=19.\begin{align*}w + 11 = 19.\end{align*}

Solve for w\begin{align*}w\end{align*} by using the Addition Property of Equality.

w+1111w=1911=8\begin{align*}w + 11 - 11 & = 19 - 11 \\ w & =8\end{align*}

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

#### Example C

In 2004, Takeru Kobayashi of Nagano, Japan, ate 5312\begin{align*}53 \frac{1}{2}\end{align*} hot dogs in 12 minutes. He broke his previous world record, set in 2002, by three hot dogs. Calculate:

a) How many minutes it took him to eat one hot dog.

b) How many hot dogs he ate per minute.

c) What his old record was.

Solution:

a) Write an equation, letting m\begin{align*}m\end{align*} represent the number of minutes to eat one hot dog: 53.5m=12\begin{align*}53.5m = 12\end{align*}.

Applying the Multiplication Property of Equality:

53.5m53.5m=1253.5=0.224 minutes\begin{align*}\frac{53.5m}{53.5} & = \frac{12}{53.5} \\ m & = 0.224\ minutes\end{align*}

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 exercises.

### Guided Practice

In a previous Concept, we calculated that Mayra could 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\begin{align*}speed=\frac{distance}{time}\end{align*}.

Substituting in speed=6.5\begin{align*}speed=6.5\end{align*} and time=2.25\begin{align*}time=2.25\end{align*} we get:

6.5=distance3.25.\begin{align*}6.5=\frac{distance}{3.25}.\end{align*}

Now we 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\begin{align*} &6.5\times 2.25=\frac{distance}{2.25}\times 2.25\\ &6.5\times 2.25=\frac{distance}{\cancel{2.25}}\times \cancel{2.25}\\ &6.5\times 2.25=distance\\ &13.5=distance\\ \end{align*}

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

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: One-Step Equations (12:30)

1. 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.
1. How many more tokens he needs to collect, n\begin{align*}n\end{align*}.
2. How many tokens he collects per week, w\begin{align*}w\end{align*}.
3. How many more weeks remain until he can send off for his boat, r\begin{align*}r\end{align*}.
2. 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.
1. The amount of money that he sells the cake for (u)\begin{align*}(u)\end{align*}.
2. The amount of money he charges for each slice (c)\begin{align*}(c)\end{align*}.
3. The total profit he makes on the cake (w)\begin{align*}(w)\end{align*}.
3. Solve the remaining two questions regarding Takeru Kobayashi in Example C.

Mixed Review

1. Simplify 48\begin{align*}\sqrt{48}\end{align*}.
2. Classify 6.23 according to the real number chart.
3. Reduce 1184\begin{align*}\frac{118}{4}\end{align*}.
4. Graph the following ordered pairs: {(2,2),(4,1),(5,5),(3,2)}\begin{align*}\left \{(2,-2),(4,-1),(5,-5),(3,-2) \right \}\end{align*}.
5. Define evaluate.
6. Underline the math verb in this sentence: m\begin{align*}m\end{align*} minus n\begin{align*}n\end{align*} is 16.
7. What property is illustrated here? 4(a+11.2)=4(a)+4(11.2)\begin{align*}4(a + 11.2) = 4(a) + 4(11.2)\end{align*}

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Color Highlighted Text Notes

### Vocabulary Language: English Spanish

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 $c$: If $a = b$, then $a(c)= b(c).$

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