SOLUTION: Unit 4a linear equations updated october 2017 2 pdf ... - Free Printable

You’ve uploaded a worksheet titled “LINEAR EQUATIONS WORD PROBLEMS”, which is divided into two types:

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## ✔ Type 1: SLOPE-INTERCEPT FORM (y = mx + b)
*Used when given a rate of change (m) and a starting point (b).*

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Problem 1: Evan at the County Fair

> Given:
> - Admission = $5 (starting cost → b)
> - Cost per ride = $0.50 (rate of change → m)
> - Total money = $20 → we set y = 20
>
> Equation:
> `y = 0.50x + 5.00`
>
> Solve for x (number of rides):
> `20 = 0.50x + 5`
> Subtract 5: `15 = 0.50x`
> Divide by 0.50: `x = 30`
>
> ✔ Answer: 30 rides

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Problem 2: Sally Kayaking at Crimson Lake

> Given:
> - Deposit = $25.00 (b)
> - Rate per hour = $8.50 (m)
> - Time rented: 11:30 a.m. to 2:30 p.m. = 3 hours → x = 3
>
> Equation:
> `y = 8.50x + 25.00`
>
> Plug in x = 3:
> `y = 8.50(3) + 25 = 25.50 + 25 = 50.50`
>
> ✔ Answer: $50.50

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Problem 3: Alex’s Truck Depreciation

> Given:
> - Initial value = $42,935 (b)
> - Depreciates $4,200/year → m = -4200 (negative because it’s decreasing)
> - Final value = $5,135 → set y = 5135
>
> Equation:
> `y = -4200x + 42935`
>
> Solve for x (years):
> `5135 = -4200x + 42935`
> Subtract 42935: `-37800 = -4200x`
> Divide by -4200: `x = 9`
>
> ✔ Answer: 9 years

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Problem 4: Car Wash and Gas Purchase

> Given:
> - Car wash = $6.00 (b)
> - Gas = $2.35/gallon (m)
> - Total budget = $40 → y = 40
>
> Equation:
> `y = 2.35x + 6.00`
>
> Solve for x (gallons):
> `40 = 2.35x + 6`
> Subtract 6: `34 = 2.35x`
> Divide by 2.35: `x ≈ 14.47`
>
> ✔ Answer: ~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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## ✔ Type 2: STANDARD FORM (Ax + By = C)
*Used when problem relates 2 different objects — often solved with substitution or elimination.*

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Problem 5: Sam’s Concession Stand Order

> Given:
> - 2 hamburgers + 3 hotdogs = $19.05
> - Hamburgers = $5.25 each
> - Let x = hamburgers, y = hotdogs
>
> Equation setup:
> `2x + 3y = 19.05`
> But since we know x = 5.25 (price per hamburger), substitute:
> `2(5.25) + 3y = 19.05`
> `10.50 + 3y = 19.05`
> Subtract 10.50: `3y = 8.55`
> Divide by 3: `y = 2.85`
>
> ✔ Answer: Each hot dog costs $2.85

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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 hot dog |

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## 🧠 Key Concepts Reinforced:

- Slope-intercept form (y = mx + b) is perfect for problems with:
- A fixed starting cost (b)
- A constant rate per unit (m)

- Standard form is useful for combining two different items with known quantities and total cost.

- Always define your variables clearly!
- Check units and make sure your answer makes sense in context.

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✔ You've successfully solved all 5 problems! Great job applying linear equations to real-world scenarios.