Lesson 25
Using Linear Relations to Solve Problems
Let’s write equations for real-world situations and think about their solutions.
25.1: Buying Fruit
For each relationship described, write an equation to represent the relationship.
- Grapes cost $2.39 per pound. Bananas cost $0.59 per pound. You have $15 to spend on \(g\) pounds of grapes and \(b\) pounds of bananas.
- A savings account has $50 in it at the start of the year and $20 is deposited each week. After \(x\) weeks, there are \(y\) dollars in the account.
25.2: Five Savings Accounts
Each line represents one person’s weekly savings account balance from the start of the year.
![Coorindate grid. Lines a, c, d, e, h.](https://staging-cms-im.s3.amazonaws.com/Zp7iT6GjkaXBZKoQyEZDwPW4?response-content-disposition=inline%3B%20filename%3D%228-8.3.D2.Image.06.png%22%3B%20filename%2A%3DUTF-8%27%278-8.3.D2.Image.06.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T051735Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c296c5e228e5cc21788fe4ba18d9d9d5feaacad71feeb23815eaccbc70b2ae28)
- Choose one line and write a description of what happens to that person's account over the first 17 weeks of the year. Do not tell your group which line you chose.
- Share your story with your group and see if anyone can guess your line.
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Write an equation for each line on the graph. What do the slope, \(m\), and vertical intercept, \(b\), in each equation mean in the situation?
- For which equation is \((1,70)\) a solution? Interpret this solution in terms of your story.
- Predict the balance in each account after 20 weeks.
25.3: Fabulous Fish
The Fabulous Fish Market orders tilapia, which costs $3 per pound, and salmon, which costs $5 per pound. The market budgets $210 to spend on this order each day.
- What are five different combinations of salmon and tilapia that the market can order?
- Define variables and write an equation representing the relationship between the amount of each fish bought and how much the market spends.
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Sketch a graph of the relationship. Label your axes.
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On your graph, plot and label the combinations A—F.
A B C D E F pounds of tilapia 5 19 27 25 65 55 pounds of salmon 36 30.6 25 27 6 4 -
Which of these combinations can the market order? Explain or show your reasoning.
- List two ways you can tell if a pair of numbers is a solution to an equation.