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# Applications Using Linear Models

## Solve story problems using linear equations

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Solve Real-World Problems by Using Strategies and a Plan
License: CC BY-NC 3.0

Cameron is going to visit his grandparents. His flight leaves at 9:00 pm. He must be at the airport two hours before departure and he lives one hour from the airport. Cameron knows that he will need one and a half hours to pack and get ready for his trip. Cameron arrives home from school at 4:00 pm. How can Cameron figure out if he will make his flight in time?

In this concept, you will learn to solve real-world problems by using strategies and a plan.

### Using Strategies to Solve Problems

When problem-solving, you will use strategies as part of a plan. For each situation, you will be asked to read and understand a given problem. You will make a plan to solve by choosing appropriate strategies. There are multiple ways to solve a word problem. Therefore, you will need to consider and compare different approaches for each problem given.

First, always read the problem completely. Do not skip words or sections since they may be key to your understanding of what the problem is asking you to do. Make note of the question being asked by highlighting it. Another tip is to underline or circle all clues that may help you with planning and strategy.

Let’s apply this strategy to an example.

A lizard ate five hundred flies on five consecutive nights. Each night he ate twenty-five more than the night before. How many flies did the lizard eat each night?

For this problem, you are to determine the number of flies the lizard ate each night. Here is what you are told:

• You are told that the lizard ate a total of five hundred flies over the course of five nights.
• You are told that the lizard eats twenty-five more flies each night than the night before.
• You should know the word consecutive means a logical sequence or succession. In this case it means one night after the other.

Next, make a plan to solve the problem. Use the information listed above to create a verbal model and use this model to write and solve an equation to answer the highlighted question.

Next, write the verbal model.

Number of flies eaten on night one + (number of flies eaten on night one + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five + twenty-five + twenty-five) = total number of flies eaten over the five nights (500).

Next, name the variable. Let “\begin{align*}x\end{align*}” represent the number of flies eaten on night one.

Next, write a single-variable equation to represent the verbal model.

\begin{align*}x+(x+25)+(x+25+25)+(x+25+25+25)+(x+25+25+25+25)=500\end{align*}

Next, simplify the equation on the left side of the equal sign by performing the addition in the parenthesis.

\begin{align*}x+(x+25)+(x+50)+(x+75)+(x+100)=500\end{align*}

There is \begin{align*}a +1\end{align*} understood as being in front of each set of parenthesis.

Next, apply the distributive property to clear the parenthesis.

\begin{align*}x+x+25+x+50+x+75+x+100=500\end{align*}

Next, simplify the left side of the equation.

\begin{align*}5x+250=500\end{align*}

Next, isolate the variable by subtracting 250 from both sides of the equation.

\begin{align*}5x+250-250=500-250\end{align*}

Next, simplify both sides of the equation.

\begin{align*}5x=250\end{align*}

Next, divide both sides of the equation by 5.

\begin{align*}\begin{array}{rcl} \frac{^1 \cancel{5}x}{\cancel{5}} & = & \frac{250}{5}\\ x & = & 50 \end{array}\end{align*}

Remember “\begin{align*}x\end{align*}” represents the number of flies the lizard ate on night one.

Next, substitute \begin{align*}x=50\end{align*} into the expression representing each night to determine the number of flies the lizard ate on each of the five consecutive nights.

Night one = \begin{align*}x = 50\end{align*}

Night two = \begin{align*}x+25=50+25=75\end{align*}

Night three = \begin{align*}x+25+25=50+25+25=100\end{align*}

Night four = \begin{align*}x+25+25+25=50+25+25+25=125\end{align*}

Night five = \begin{align*}x+25+25+25+25=50+25+25+25+25=150\end{align*}

Then, check to determine if the solutions make a true statement. Substitute \begin{align*}x=50\end{align*} into the original equation.

\begin{align*}\begin{array}{rcl} && 50 + (50 + 25)+(50+25+25)+ (50 + 25+25+25)+ (50 + 25+25+25+25)=500\\ && 50 + 75 + 100 + 125 + 150=500\\ && 500 = 500 \end{array}\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Cameron trying to figure out if he will get to the airport on time. This is his timeline:

• Cameron arrived home at 4:00 pm.
• He needs one and a half hours to pack.
• He has a one hour drive to get to the airport.
• Cameron must be at the airport two hours prior to his departure time.
• His flight leaves at 9:00 pm.

Next, Cameron has to make a plan of what he can do with all of this information.

Working backwards from the departure time will tell him what time he should leave the house.

\begin{align*}\begin{array}{rcl} && \text{Departure Time} - 2 \text{ hours} = \text{Check-In Time}\\ && 9:00 \text{ pm} - 2:00 \text{ hours} = 7:00 \text{ pm} \\ \\ && \text{Check-In Time} - 1 \text{ hour drive} = \text{Leave home Time}\\ && 7:00 \text{ pm} - 1:00 \text{ pm} = 6:00 \text{ pm} \\ \\ && \text{Leave Home Time} - \text{Pack Time} = \text{Time to arrive home}\\ && 6:00 \text{ pm} - 1\frac{1}{2} \text{ hours} = 4:30 \text{ pm}\\ && 6:00 \text{ pm} - 1:30 = 4:30 \text{ pm} \end{array}\end{align*}

Cameron arrived home at 4:00 pm so he should be able to make his flight.

#### Example 2

A train’s caboose is 12 feet long. Each of the train’s eight cars are twice the length of the caboose. What is the length of the entire train?

First, read the problem. Highlight the question and underline key words that will help you to answer the question being asked.

Next, confirm what the problem is asking you to do and list the information given in the problem.

You are asked to determine the length of the entire train. You were told some information.

• The caboose is 12 feet long.
• There are an additional eight cars.
• Each car is twice the length of the caboose.

Next, create a plan for solving the problem.

You can draw a diagram and use a verbal model to visualize the information given in the problem. Then, write an equation to determine the length of the entire train.

License: CC BY-NC 3.0

Verbal Model: Eight trains twice the length of the caboose + the length of the caboose = length of the entire train

Next, name the variable. Let \begin{align*}x\end{align*}” represent the length of the entire train.

Next, write and solve a single-variable equation to represent the verbal model.

\begin{align*}8(2 \cdot 12) + 12 = x\end{align*}

First, perform the multiplication is the parenthesis \begin{align*}(2 \cdot 12)=24\end{align*} and write the new equation.

\begin{align*}8(24) +12=x\end{align*}

Next, multiply: \begin{align*}8(24)=192\end{align*} to clear the parenthesis. Write the new equation.

\begin{align*}192+12=x\end{align*}

Then, add the numbers on the left side of the equation. \begin{align*}192 + 12 =204\end{align*}

\begin{align*}204 = x\end{align*}

The answer is 204.

The length of the entire train is 204 feet.

#### Example 3

A ten year old's heart beats approximately 85 times per minute. How many times does the heart beat in 24 seconds?

First, read the problem. Highlight the question and underline key words that will help you to answer the question being asked.

Next, confirm what the problem is asking you to do and list the information given in the problem.

You are asked to determine the number of heart beats in 24 seconds. You were told some information.

• The heart beats 85 times per minute.

Next, create a plan for solving the problem.

You are being asked to find an unknown rate (number of heart beats in 24 seconds) which tells you to write a proportion to solve the problem.

Next, name the variable in the proportion.

Let “\begin{align*}x\end{align*}” represent the number of heartbeats in 24 seconds.

Next, write and solve the proportion. Remember, there are 60 seconds in one minute.

First, write down what you know \begin{align*}(85 \text{ beats} = 60 \text{ seconds})\end{align*} and express it over what you want to know \begin{align*}(x=24 \text{ seconds})\end{align*}.

\begin{align*}\frac{85 \text{ beats}}{x} = \frac{60 \text{ seconds}}{24 \text{ seconds}}\end{align*}

Next, cross out the seconds on the right side of the proportion. Write the new proportion.

\begin{align*}\begin{array}{rcl} \frac{85 \text{ beats}}{x} & = & \frac{60 \cancel{\text{ seconds}}}{24\cancel{\text{ seconds}}}\\ \frac{85 \text{ beats}}{x} & = & \frac{60}{24} \end{array}\end{align*}

Next, multiply both sides of the proportion by \begin{align*}24x\end{align*} and write the new equation.

\begin{align*}\begin{array}{rcl} 24 \cancel{x}\left ( \frac{85 \text{ beats}}{\cancel{x}} \right ) & = & \cancel{24} x\left ( \frac{60}{\cancel{24}} \right )\\ 2040 & = & 60x \end{array}\end{align*}

Then, divide both sides of the equation by 60.

\begin{align*}\begin{array}{rcl} \frac{2040 \text{ beats}}{60} & = & \frac{^1\cancel{60}x}{\cancel{60}}\\ 34 \text{ beats} & = & x \end{array}\end{align*}

The answer is 34.

The answer is \begin{align*}x = 34\end{align*} beats in 24 seconds.

### Review

Read each problem and then answer the questions below the problem.

Ted has a collection of rare coins. He already had 34 coins in his collection. The first week, Ted purchases 1 new coin. During the second week, Ted purchases 4 coins. During the third week, Ted adds 9 new coins to his collection. At this rate how many weeks will it take Ted to collect 125 coins?

1. Which strategy should Ted use to solve this problem?

2. What could Ted draw to help him with his solution?

3. How long will it take Ted to collect 125 coins?

Savannah wants to buy a pair of jeans that cost $59.00. They are on sale for 25% off. 4. Which strategy could Savannah use to calculate the price? 5. What is the amount of the discount? 6. What is the sale price? Carlos was in charge of organizing cookies for a bake sale. He organized them into bundles of six cookies. When he was done, he had 15 bundles of cookies. How many cookies did he start with? 7. Which strategy could you use to solve this problem? 8. Write an equation to describe the problem. 9. Solve the equation. 10. How many cookies did Carlos begin with? Veronica made brownies. She made twice as many brownies as Carlos had cookies. How many brownies did she make? 11. Which strategy could you use to solve this problem? 12. Write an equation to describe the problem. 13. Solve the equation. 14. How many brownies did Veronica make? If Veronica sold half of the cookies that she made, how many would be sold? If she charged$1.50 per brownie, how much money would she make?

15. Which strategy could you use to solve this problem?

16. How much money would she make?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.16.

### Notes/Highlights Having trouble? Report an issue.

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### Vocabulary Language: English

Proportion

A proportion is an equation that shows two equivalent ratios.

### Image Attributions

1. [1]^ License: CC BY-NC 3.0
2. [2]^ License: CC BY-NC 3.0

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