Worksheet with six word problems on linear equations (Y = mx + b), covering real-world applications like water level changes, movie pass costs, babysitting fees, plumber charges, recycling, and canoe rentals.

Let's solve each of these slope-intercept form word problems step by step using the equation:

$$
y = mx + b
$$

Where:
- $ y $ is the dependent variable (output),
- $ x $ is the independent variable (input),
- $ m $ is the slope (rate of change),
- $ b $ is the y-intercept (initial value).

---

Problem 1: River Water Level

> Suppose that the water level of a river is 34 feet and that it is receding at a rate of 0.5 foot per day. Write an equation for the water level, $ L $, after $ d $ days. In how many days will the water level be 26 feet?

#### Step 1: Identify variables
- Initial water level = 34 feet → this is the y-intercept ($ b $)
- Rate of change = -0.5 ft/day (receding) → slope ($ m $)
- Let $ L $ = water level after $ d $ days

So,
$$
L = -0.5d + 34
$$

#### Step 2: Find when $ L = 26 $
$$
26 = -0.5d + 34
$$
Subtract 34 from both sides:
$$
-8 = -0.5d
$$
Divide both sides by -0.5:
$$
d = \frac{-8}{-0.5} = 16
$$

✔ Answer: The water level will be 26 feet after 16 days.

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Problem 2: Movie Pass

> Seth’s father is thinking of buying his son a six-month movie pass for $40. With the pass, matinees cost $1.00. If matinees are normally $3.50 each, how many times must Seth attend in order for it to benefit his father to buy the pass?

#### Step 1: Define variables
Let $ x $ = number of movies attended

- Without pass: Cost = $ 3.50x $
- With pass: Cost = $ 40 + 1.00x $

We want to find when with pass < without pass:
$$
40 + 1x < 3.50x
$$

Subtract $ 1x $ from both sides:
$$
40 < 2.50x
$$

Divide both sides by 2.50:
$$
x > \frac{40}{2.50} = 16
$$

So, Seth must attend more than 16 times for the pass to be beneficial.

Since he must attend at least 17 times, the break-even point is 17.

✔ Answer: Seth must attend 17 times or more for the pass to be worth it.

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Problem 3: Babysitting

> Nicole charges a flat fee of $3, plus $5 per hour. Write an equation for the cost, $ C $, after $ h $ hours of babysitting. What do you think the slope and the y-intercept represent? How much money will she make if she babysits 5 hours?

#### Step 1: Equation
- Flat fee = $3 → y-intercept ($ b $)
- $5 per hour → slope ($ m $)

So:
$$
C = 5h + 3
$$

#### Interpretation:
- Slope (5): Amount earned per hour ($5/hour)
- Y-intercept (3): Base fee charged even if no time is spent (flat fee)

#### Step 2: For 5 hours:
$$
C = 5(5) + 3 = 25 + 3 = 28
$$

✔ Answer:
- Equation: $ C = 5h + 3 $
- Slope: $5/hour, y-intercept: $3 flat fee
- Earnings for 5 hours: $28

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Problem 4: Plumber Charges

> A plumber charges $25 for a service call plus $50 per hour of service. Write an equation in slope-intercept form for the cost, $ C $, after $ h $ hours of service. What will be the total cost for 8 hours of work? 10 hours of work?

#### Step 1: Equation
- Service call fee = $25 → y-intercept
- $50 per hour → slope

So:
$$
C = 50h + 25
$$

#### Step 2: Cost for 8 hours:
$$
C = 50(8) + 25 = 400 + 25 = 425
$$

#### Step 3: Cost for 10 hours:
$$
C = 50(10) + 25 = 500 + 25 = 525
$$

✔ Answer:
- Equation: $ C = 50h + 25 $
- 8 hours: $425
- 10 hours: $525

---

Problem 5: Aluminum Cans

> Rufus collected 100 pounds of aluminum cans to recycle. He plans to collect an additional 25 pounds each week. Write and graph the equation for the total pounds, $ P $, of aluminum cans after $ w $ weeks. What does the slope and y-intercept represent? How long will it take Rufus to collect 400 pounds of cans?

#### Step 1: Equation
- Initial amount = 100 lbs → y-intercept
- Add 25 lbs/week → slope

So:
$$
P = 25w + 100
$$

#### Interpretation:
- Slope (25): 25 pounds collected per week
- Y-intercept (100): Starting amount before any weeks passed

#### Step 2: When $ P = 400 $
$$
400 = 25w + 100
$$
$$
300 = 25w
$$
$$
w = \frac{300}{25} = 12
$$

✔ Answer:
- Equation: $ P = 25w + 100 $
- Slope: 25 lbs/week, y-intercept: 100 lbs initial
- Time to reach 400 lbs: 12 weeks

> 📌 Note: To graph, plot points like (0,100), (1,125), (2,150), etc., and draw a line.

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Problem 6: Canoe Rental

> A canoe rental service charges a $20 transportation fee and $30 dollars an hour to rent a canoe. Write and graph an equation representing the cost, $ y $, of renting a canoe for $ x $ hours. What is the cost of renting the canoe for 6 hours?

#### Step 1: Equation
- Transportation fee = $20 → y-intercept
- $30/hour → slope

So:
$$
y = 30x + 20
$$

#### Step 2: Cost for 6 hours:
$$
y = 30(6) + 20 = 180 + 20 = 200
$$

✔ Answer:
- Equation: $ y = 30x + 20 $
- Graph: Start at (0,20), go up 30 units per 1 unit right.
- Cost for 6 hours: $200

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✔ Final Summary of Answers:

| Problem | Answer |
|--------|--------|
| 1 | Equation: $ L = -0.5d + 34 $; Water level reaches 26 ft in 16 days |
| 2 | Must attend 17 times for the pass to be beneficial |
| 3 | Equation: $ C = 5h + 3 $; Slope = $5/hour, y-int = $3 flat fee; Earns $28 for 5 hours |
| 4 | Equation: $ C = 50h + 25 $; Cost for 8 hrs = $425, 10 hrs = $525 |
| 5 | Equation: $ P = 25w + 100 $; Slope = 25 lbs/week, y-int = 100 lbs; Takes 12 weeks to reach 400 lbs |
| 6 | Equation: $ y = 30x + 20 $; Cost for 6 hours = $200 |

Let me know if you'd like graphs drawn for Problems 5 or 6!