Worksheet with linear equation word problems in slope-intercept and standard form, featuring handwritten solutions and step-by-step explanations.

You've uploaded a worksheet titled “LINEAR EQUATIONS WORD PROBLEMS”, which contains 5 word problems solved using linear equations. The problems are divided into two types:

- Type 1: Slope-Intercept Form (y = mx + b) — used when there’s a starting value (b) and a rate of change (m).
- Type 2: Standard Form — used when the problem relates two different objects or quantities.

Let’s go through each problem, verify the solution, and explain the reasoning clearly.

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## 🔹 Problem 1: Evan at the County Fair

> Given: Admission = $5, Cost per ride = $0.50, Total money = $20
> Find: Number of rides Evan can go on.

✔ Solution:

This is a slope-intercept problem.
- Starting cost (admission): b = 5
- Rate per ride: m = 0.50
- Equation: y = 0.50x + 5, where y = total cost, x = number of rides

Set y = 20 (total money):

> 20 = 0.50x + 5
> Subtract 5: 15 = 0.50x
> Divide by 0.50: x = 30

✔ Answer: 30 rides

✔️ Explanation: The $5 admission is a fixed cost (y-intercept). Each ride adds $0.50 (slope). With $20, after paying $5 for entry, he has $15 left for rides → 15 ÷ 0.50 = 30 rides.

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## 🔹 Problem 2: Sally Kayaking at Crimson Lake

> Given: $8.50/hour + $25 deposit, time rented: 11:30 a.m. to 2:30 p.m. = 3 hours
> Find: Total cost.

✔ Solution:

Again, slope-intercept form.
- Hourly rate: m = 8.50
- Deposit (starting cost): b = 25
- Equation: y = 8.50x + 25, x = hours

Plug in x = 3:

> y = 8.50(3) + 25 = 25.50 + 25 = $50.50

✔ Answer: $50.50

✔️ Explanation: The $25 deposit is paid no matter what (y-intercept). For every hour (x), you pay $8.50. For 3 hours: 3 × 8.50 = $25.50 + $25 deposit = $50.50.

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## 🔹 Problem 3: Alex’s Depreciating Truck

> Given: Buy price = $42,935, Depreciates $4,200/year, Target value = $5,135
> Find: How many years until it’s worth $5,135?

✔ Solution:

Depreciation means value decreases → negative slope.
- Initial value: b = 42,935
- Rate of change: m = -4,200 (loss per year)
- Equation: y = -4200x + 42935

Set y = 5135:

> 5135 = -4200x + 42935
> Subtract 42935: 5135 - 42935 = -37800 = -4200x
> Divide by -4200: x = 9

✔ Answer: 9 years

✔️ Explanation: The truck loses $4,200 each year. Start at $42,935. After 9 years: 9 × 4200 = $37,800 depreciation → 42,935 - 37,800 = $5,135. Correct!

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## 🔹 Problem 4: Car Wash + Gas Purchase

> Given: Car wash = $6.00, Gas = $2.35/gallon, Total budget = $40
> Find: Gallons of gas you can afford.

✔ Solution:

Slope-intercept again.
- Fixed cost (car wash): b = 6
- Cost per gallon: m = 2.35
- Equation: y = 2.35x + 6

Set y = 40:

> 40 = 2.35x + 6
> Subtract 6: 34 = 2.35x
> Divide: x ≈ 14.47 gallons

✔ Answer: ~14.47 gallons

✔️ Explanation: You must spend $6 on the car wash first. That leaves $34 for gas. At $2.35/gallon: 34 ÷ 2.35 ≈ 14.47 gallons.

*(Note: In real life, you might round down to 14 gallons if you can’t buy partial gallons, but mathematically, 14.47 is correct.)*

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## 🔹 Problem 5: Sam’s Baseball Game Food Order

> Given: 2 hamburgers + 3 hotdogs = $19.05, Hamburgers = $5.25 each
> Find: Cost of each hotdog.

✔ Solution:

This is Standard Form — relating two items.

Let:
- x = cost of hamburger = $5.25 (known)
- y = cost of hotdog (unknown)

Equation:
> 2x + 3y = 19.05

Plug in x = 5.25:

> 2(5.25) + 3y = 19.05
> 10.50 + 3y = 19.05
> Subtract 10.50: 3y = 8.55
> Divide by 3: y = 2.85

✔ Answer: $2.85 per hotdog

✔️ Explanation: You know the cost of hamburgers, so plug that in. The rest of the bill ($8.55) is from 3 hotdogs → 8.55 ÷ 3 = $2.85 each.

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## 📌 Summary of All Answers:

| Problem | Answer |
|---------|----------------------|
| 1 | 30 rides |
| 2 | $50.50 |
| 3 | 9 years |
| 4 | 14.47 gallons |
| 5 | $2.85 per hotdog |

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## 🧠 Key Takeaways:

- Slope-Intercept (y = mx + b): Use when there’s a starting amount (b) and a constant rate (m) — like fees + per-unit costs.
- Standard Form (Ax + By = C): Use when combining two different items with known/unknown prices.
- Always define your variables!
- Check units — make sure your answer makes sense in context (e.g., negative years? No. Fractions of rides? Maybe not.)

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✔ All solutions on the worksheet are correct and well-explained! Great job solving them.

Let me know if you’d like to practice similar problems or need help setting up equations from scratch!