Writing Linear Equations (y=mx+b) - Pre Algebra

1. A candle burns at a constant rate. Initially it is 8 inches tall. After burning for 3 hours, it is 6.5 inches tall. After burning for 7 hours, it is 4.5 inches tall. Which equation models the candle's height $y$ (in inches) after burning for $x$ hours?
   Answer: $y = -0.5x + 8$ Using points (3, 6.5) and (7, 4.5), the slope is $m = \frac{4.5-6.5}{7-3} = \frac{-2}{4} = -0.5$. Since the candle starts at 8 inches when $x = 0$, the y-intercept is 8. Therefore, $y = -0.5x + 8$. Verify: at $x = 3$, $y = -0.5(3) + 8 = 6.5$ ✓; at $x = 7$, $y = -0.5(7) + 8 = 4.5$ ✓. The candle burns 0.5 inches per hour. Other choices give incorrect heights when substituted.
2. A water tank initially contains 150 gallons and drains at a constant rate. After 4 hours, it contains 110 gallons. After 9 hours, it contains 60 gallons. Which equation models the amount of water $y$ (in gallons) remaining after $x$ hours?
   Answer: $y = -10x + 150$ Using points (4,110) and (9,60), the slope is $m = \frac{60-110}{9-4} = \frac{-50}{5} = -10$. Using point-slope form with (4,110): $y - 110 = -10(x - 4)$, which gives $y = -10x + 150$. This means the tank starts with 150 gallons (when $x = 0$) and drains 10 gallons per hour. Verify: at $x = 4$, $y = -10(4) + 150 = 110$ ✓; at $x = 9$, $y = -10(9) + 150 = 60$ ✓. The other choices give incorrect values when substituted.
3. A parking garage charges a flat entrance fee plus an hourly rate. The total cost for parking 3 hours is $12, and the total cost for parking 7 hours is $20. If $y$ represents the total cost and $x$ represents the number of hours, which equation represents this situation?
   Answer: $y = 2x + 6$ First find the slope: $m = \frac{20-12}{7-3} = \frac{8}{4} = 2$. Then use point-slope form with (3,12): $y - 12 = 2(x - 3)$, which simplifies to $y = 2x + 6$. We can verify: when $x = 3$, $y = 2(3) + 6 = 12$ ✓, and when $x = 7$, $y = 2(7) + 6 = 20$ ✓. Choice B gives incorrect values ($y = 3(3) + 3 = 12$ works but $y = 3(7) + 3 = 24 \neq 20$). Choice C gives $y = 4(3) = 12$ but $y = 4(7) = 28 \neq 20$. Choice D gives $y = 6(3) - 6 = 12$ but $y = 6(7) - 6 = 36 \neq 20$.
4. A delivery service charges based on package weight. For a 5-pound package, the cost is $18. For a 12-pound package, the cost is $32. Assuming the relationship is linear, which equation represents the cost $y$ for a package weighing $x$ pounds?
   Answer: $y = 2x + 8$ Using points (5,18) and (12,32), the slope is $m = \frac{32-18}{12-5} = \frac{14}{7} = 2$. Using point-slope form with (5,18): $y - 18 = 2(x - 5)$, which gives $y = 2x + 8$. This means $8 base charge plus $2 per pound. Verify: for 12 pounds, $y = 2(12) + 8 = 32$ ✓. Choice B gives $y = 3(5) + 3 = 18$ ✓ but $y = 3(12) + 3 = 39 \neq 32$. Choice C gives $y = 1.5(5) + 10.5 = 18$ ✓ but $y = 1.5(12) + 10.5 = 28.5 \neq 32$. Choice D gives $y = 2.5(5) + 5.5 = 18$ ✓ but $y = 2.5(12) + 5.5 = 35.5 \neq 32$.

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