Slope-Intercept Form Word Problems worksheet with three real-world scenarios.

Let's solve each problem step by step using slope-intercept form, which is:

$$
y = mx + b
$$

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

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Problem 1: John collects aluminum cans

Given:
- Initial amount: 100 pounds
- Additional: 25 pounds per week

#### (a) Write a linear equation in slope-intercept form

Let:
- $ y $ = total weight of cans (in pounds)
- $ x $ = number of weeks

The rate of change (slope) is 25 pounds per week.
The initial amount (y-intercept) is 100 pounds.

So the equation is:

$$
y = 25x + 100
$$

✔ Answer (a): $ y = 25x + 100 $

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#### (b) In how many weeks will John have a total of 400 pounds?

Set $ y = 400 $ and solve for $ x $:

$$
400 = 25x + 100
$$

Subtract 100 from both sides:

$$
300 = 25x
$$

Divide by 25:

$$
x = \frac{300}{25} = 12
$$

✔ Answer (b): 12 weeks

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Problem 2: Kris’s plant growth

Given:
- Initial height: 5 cm
- Growth rate: 1.5 cm per day

#### (a) Write an equation for the plant's height after $ x $ days

Let:
- $ y $ = height of plant (cm)
- $ x $ = number of days

Slope $ m = 1.5 $, y-intercept $ b = 5 $

$$
y = 1.5x + 5
$$

✔ Answer (a): $ y = 1.5x + 5 $

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#### (b) How tall will the plant be after 9 days?

Plug $ x = 9 $ into the equation:

$$
y = 1.5(9) + 5 = 13.5 + 5 = 18.5
$$

✔ Answer (b): 18.5 cm

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Problem 3: Plane descending

Given:
- Maximum altitude: 7500 meters
- Rate of descent: 5 meters per second

Note: Since it's losing altitude, the slope is negative.

#### (a) Write an equation for altitude after $ x $ seconds

Let:
- $ y $ = altitude (meters)
- $ x $ = time in seconds

Slope $ m = -5 $ (descending), y-intercept $ b = 7500 $ (starting at max altitude)

$$
y = -5x + 7500
$$

✔ Answer (a): $ y = -5x + 7500 $

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#### (b) Find the time it takes to meet the ground

When the plane reaches the ground, $ y = 0 $. Solve for $ x $:

$$
0 = -5x + 7500
$$

Add $ 5x $ to both sides:

$$
5x = 7500
$$

Divide by 5:

$$
x = \frac{7500}{5} = 1500
$$

✔ Answer (b): 1500 seconds

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✔ Final Answers Summary:

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1. John's Aluminum Cans

(a) $ y = 25x + 100 $
(b) 12 weeks

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2. Kris's Plant

(a) $ y = 1.5x + 5 $
(b) 18.5 cm

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3. Descending Plane

(a) $ y = -5x + 7500 $
(b) 1500 seconds

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Let me know if you'd like these problems graphed or explained further!