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Math problem set Grade 8 Math

Harlow Reach New Well Financing: A System of Equations Break-Even

A grade 8 problem set where students write and solve a system of two linear cost equations to find the break-even year for a fictional town's new well.

Objectives

  1. Students will write two linear equations that model competing cost plans.
  2. Students will solve a system of two linear equations to find a break-even point.

Vocabulary

Key words: system of equations, break-even, linear equation

Teacher overview

Harlow Reach is a fictional town that needs a new drinking-water well and is choosing between two contractor payment plans. One plan charges more at the start but less each year; the other charges less at the start but more each year. Writing each plan as a linear equation and solving the system shows the year the two plans cost the same amount. A common mistake is comparing only the starting amounts and ignoring the yearly rate, so students should compare the full cost over several years, not just the upfront figure.

Harlow Reach New Well Financing: A System of Equations Break-Even — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

The fictional town of Harlow Reach is comparing two well-financing plans. Let y stand for the number of years the well is used, and let C stand for the total cost in dollars.

  • Plan A costs $360,000 up front plus $15,000 each year.
  • Plan B costs $180,000 up front plus $45,000 each year.
  1. Write an equation for Plan A's total cost, C, after y years.
  2. Write an equation for Plan B's total cost, C, after y years.
  3. Find Plan A's total cost after 4 years.
  4. Find Plan B's total cost after 4 years.
  5. Set the two equations equal and solve for y, the number of years where the two plans cost the same.
  6. What is the total cost of each plan at the break-even year from item 5?
  7. Harlow Reach expects to use the well for 10 years. Which plan costs less over 10 years, and by how much?
  8. Suppose the town only needs the well for 3 years. Which plan costs less over 3 years, and by how much?
  9. Explain your reasoning: What does the break-even year tell the town about which plan to choose?

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: C = 360,000 + 15,000y.
    Plan A starts at $360,000 and adds $15,000 for each year y, so the total is 360,000 + 15,000y.
  2. Answer: C = 180,000 + 45,000y.
    Plan B starts at $180,000 and adds $45,000 for each year y, so the total is 180,000 + 45,000y.
  3. Answer: $420,000.
    C = 360,000 + 15,000(4) = 360,000 + 60,000 = 420,000 dollars.
  4. Answer: $360,000.
    C = 180,000 + 45,000(4) = 180,000 + 180,000 = 360,000 dollars.
  5. Answer: 6 years.
    Set 360,000 + 15,000y = 180,000 + 45,000y. Subtract 15,000y from both sides: 360,000 = 180,000 + 30,000y. Subtract 180,000: 180,000 = 30,000y, so y = 6 years.
  6. Answer: $450,000.
    Plan A: 360,000 + 15,000(6) = 360,000 + 90,000 = 450,000. Plan B: 180,000 + 45,000(6) = 180,000 + 270,000 = 450,000. Both cost $450,000.
  7. Answer: Plan A by $120,000.
    Plan A: 360,000 + 15,000(10) = 510,000. Plan B: 180,000 + 45,000(10) = 630,000. Plan A costs 630,000 - 510,000 = 120,000 dollars less over 10 years.
  8. Answer: Plan B by $90,000.
    Plan A: 360,000 + 15,000(3) = 405,000. Plan B: 180,000 + 45,000(3) = 315,000. Plan B costs 405,000 - 315,000 = 90,000 dollars less over 3 years.
  9. Answer: The break-even year is the point where both plans cost the same. Before 6 years Plan B is cheaper, and after 6 years Plan A is cheaper, so the town should pick the plan that fits how long it expects to use the well.

Differentiation

Support

Provide the two equations already written for items 1 and 2 so students can start at item 3, and give a filled example of substituting a year value into a cost equation before item 3. Allow a calculator throughout.

Extension

Ask students to graph both cost lines on the same axes, mark the point where they cross, and explain how the crossing point matches the break-even year they found by solving the system.

ELL supports

Pre-teach "up front," "per year," and "break-even" with a simple two-column chart comparing the plans. Provide a sentence frame for item 9: "Before year six, Plan ___ is cheaper; after year six, Plan ___ is cheaper."