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Math problem set Grade 8 Math

Kirtlebridge Service Lines: Two Systems of Equations

A grade 8 problem set where students write and solve two systems for a fictional district's lead service line work, one for cost and one for two crews' progress.

Objectives

  1. Students will write a system of two linear equations from a stated count and a stated budget.
  2. Students will solve a system by substitution and check the solution in both equations.
  3. Students will explain what the intersection point of two progress lines means.

Vocabulary

Service line
A service line is the small pipe that connects the water main under the street to the plumbing inside one home or building. It is the last leg of the trip water makes to your tap.

Also: system of equations, substitution, intersection

Teacher overview

Federal rules now require water systems to replace nearly all lead service lines within ten years, which turns a plumbing problem into a scheduling and budgeting problem for every utility that still has them. The fictional Kirtlebridge Water District has counted its remaining lead lines, priced the short ones and the long ones separately, and put two crews on the work, and each of those situations produces a pair of equations with a single meaningful solution. The error to watch for is solving only the count equation and reporting a number of lines without checking it against the cost equation, since a system is satisfied only when both equations hold at once.

Kirtlebridge Service Lines: Two Systems of Equations — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

The fictional Kirtlebridge Water District will replace 180 lead service lines. A short line costs $4,000 to replace and a long line costs $7,000. The district has budgeted exactly $900,000 for the work. Let s stand for the number of short lines and l for the number of long lines.

  1. Write a system of two equations, one for the number of lines and one for the total cost.
  2. Solve the system to find the number of long lines the district can replace.
  3. Solve the system to find the number of short lines the district can replace.
  4. Check the answer by finding the total cost of 120 short lines at $4,000 each and 60 long lines at $7,000 each.
  5. Suppose the budget rises to $960,000 for the same 180 lines at the same prices. How many long lines can the district replace then?
  6. The day crew has already replaced 24 lines and replaces 3 more each day. The night crew has replaced 8 lines and replaces 5 more each day. After how many days will the two crews have replaced the same number of lines?
  7. The day crew started with 24 lines already replaced and replaces 3 more each day. How many lines has the day crew replaced in total after 8 days?
  8. The day crew's total is graphed as y = 24 + 3d and the night crew's total as y = 8 + 5d, where d is the number of days. Explain what the point where the two lines cross means for the district.
  9. Suppose the district decided to replace all 180 lines as long lines at $7,000 each. What would the total cost be?
  10. A clerk writes the system s + l = 180 and 2s + 2l = 400. Explain why no pair of numbers can satisfy both equations.

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: s + l = 180 and 4000s + 7000l = 900000. The first equation counts the lines and the second adds up their costs, using $4,000 for each short line and $7,000 for each long line.
  2. Answer: 60 long lines. Solve the first equation for s to get s = 180 - l, then substitute:
    4000(180 - l) + 7000l = 900000. That gives 720000 - 4000l + 7000l = 900000, so 720000 + 3000l = 900000. Subtract 720000 from both sides to get 3000l = 180000, then divide by 3000 to get l = 60.
  3. Answer: 120 short lines. Substitute l = 60 into s + l = 180 to get s = 180 - 60, which is 120 short lines.
  4. Answer: $900,000. Multiply 120 by $4,000 to get $480,000 and 60 by $7,000 to get $420,000, then add to get $900,000, which matches the budget in the second equation.
  5. Answer: 80 long lines. The count equation is unchanged, so s = 180 - l, and the cost equation becomes 4000(180 - l) + 7000l = 960000. That gives 720000 + 3000l = 960000, so 3000l = 240000 and l = 80. The district would replace 100 short lines and 80 long lines.
  6. Answer: 8 days. Write each crew's total as an equation in days d: the day crew has 24 + 3d and the night crew has 8 + 5d. Set them equal: 24 + 3d = 8 + 5d. Subtract 3d from both sides to get 24 = 8 + 2d, subtract 8 to get 16 = 2d, then divide by 2 to get d = 8.
  7. Answer: 48 lines. Multiply the daily rate by the days, 3 times 8, to get 24, then add the 24 lines already finished to get 48 lines. The night crew reaches the same total, since 8 + 5 times 8 is 8 + 40, or 48.
  8. Answer: It is the day on which both crews have replaced the same number of lines. Before that day the day crew is ahead because of its head start, and after it the night crew is ahead because it works faster.
  9. Answer: $1,260,000. Multiply 180 lines by $7,000 for each line to get $1,260,000, which is more than either budget in this set.
  10. Answer: No solution. Dividing the second equation by 2 gives s + l = 200, which contradicts the first equation s + l = 180. The same two numbers cannot add to 180 and to 200, so the two lines are parallel and never meet.

Differentiation

Support

Give these students the substitution step already set up for problems 2 and 5, so they begin from the equation in one variable. Provide a two-row organizer labeled "how many" and "how much" to sort which given number belongs in which equation, and allow a calculator throughout.

Extension

Ask students to find the largest number of long lines the fictional district could replace if the budget stayed at $900,000 but the total number of lines dropped to 150, then write a short paragraph on how reducing the count changes the mix. Have them graph both cost lines to show the answer.

ELL supports

Post a labeled drawing of a service line running from the main to a house, with "short line" and "long line" marked, so the two prices attach to a picture rather than to words alone. Provide the sentence frames "The number of short lines plus the number of long lines equals ___" and "The cost of the short lines plus the cost of the long lines equals ___."

Extensions

Have students find out whether their own state publishes a lead service line inventory, and if it does, use the count for one community to write a cost equation using the two prices from this set, noting clearly that the prices are from a fictional district rather than from the real inventory.