Y = mx + b Word Problems | Study notes Business | Docsity - Free Printable

Let’s solve each problem one by one, step by step. We’ll use the formula y = mx + b, where:

- m is the rate of change (slope)
- b is the starting value (y-intercept)
- x is the input variable (like days, hours, weeks)
- y is the output (like water level, cost, pounds)

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Problem 1:

Water level starts at 34 feet and goes down 0.5 foot per day.

We want an equation for water level L after d days.

Since it’s *receding* (going down), the slope is negative: -0.5

Starting point (when d=0): L = 34 → so b = 34

Equation:
L = -0.5d + 34

Now, when will L = 26?

Set up:
26 = -0.5d + 34

Subtract 34 from both sides:
26 - 34 = -0.5d
→ -8 = -0.5d

Divide both sides by -0.5:
d = (-8) / (-0.5) = 16

✔ So, in 16 days, water level will be 26 feet.

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Problem 2:

Movie pass costs $40 upfront. With pass, matinees are $1 each. Without pass, they’re $3.50 each.

We need to find how many times Seth must go so that buying the pass saves money.

Let x = number of matinees.

Cost with pass: 40 + 1x
Cost without pass: 3.50x

We want:
40 + x < 3.50x

Solve:

40 < 3.50x - x
40 < 2.50x

Divide both sides by 2.50:
x > 40 / 2.50
x > 16

So Seth must attend more than 16 times → meaning 17 or more.

But since we’re asked “how many times must Seth attend in order for it to benefit”, the smallest whole number greater than 16 is 17.

✔ Answer: 17 times

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Problem 3:

Nicole charges $3 flat fee + $5 per hour.

Let C = total cost, h = hours.

Equation:
C = 5h + 3

- Slope (5) = hourly rate → how much she earns per hour
- Y-intercept (3) = flat fee → what she gets even if h=0

If she babysits 5 hours:

C = 5(5) + 3 = 25 + 3 = $28

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

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Problem 4:

Plumber: $25 service call + $50 per hour.

Let C = total cost, h = hours.

Equation:
C = 50h + 25

For 8 hours:
C = 50(8) + 25 = 400 + 25 = $425

For 10 hours:
C = 50(10) + 25 = 500 + 25 = $525

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

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Problem 5:

Rufus has 100 lbs already, collects 25 lbs each week.

Let P = total pounds, w = weeks.

Equation:
P = 25w + 100

- Slope (25) = pounds collected per week
- Y-intercept (100) = starting amount

When will P = 400?

400 = 25w + 100

Subtract 100:
300 = 25w

Divide by 25:
w = 12

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

*(Note: Graphing isn’t required here since you didn’t ask for a graph — but if needed, plot points like (0,100), (1,125), (2,150)...)*

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Problem 6:

Canoe rental: $20 transportation fee + $30 per hour.

Let y = total cost, x = hours.

Equation:
y = 30x + 20

For 6 hours:
y = 30(6) + 20 = 180 + 20 = $200

✔ Equation: y = 30x + 20
Cost for 6 hours: $200

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Final Answer:
1. Equation: L = -0.5d + 34; Days to reach 26 ft: 16
2. Times to benefit: 17
3. Equation: C = 5h + 3; Slope = $5/hr, y-int = $3; Cost for 5 hrs: $28
4. Equation: C = 50h + 25; 8 hrs: $425; 10 hrs: $525
5. Equation: P = 25w + 100; Slope = 25 lbs/wk, y-int = 100 lbs; Time to 400 lbs: 12 weeks
6. Equation: y = 30x + 20; Cost for 6 hrs: $200