Math problem set Grade 8 Math

Hartsell Station Two Tanks: A Systems of Equations Problem Set

A grade 8 problem set where students model two tanks, one draining and one filling, as linear equations and solve the system to find when their depths are equal.

Grade: Grade 8
Subject: Math
Material: Math problem set
Time: 45 min
Prep: No prep
Materials: Calculator, Graph paper

Standards

8.EE.C.8 Common Core Math

Analyze and solve pairs of simultaneous linear equations, understanding that solutions correspond to points of intersection, and solving systems by graphing and algebraically in the context of real-world problems.

How this material aligns: Students write two linear equations for the tank depths, build a table, and solve the system algebraically in problems 3 through 6, finding the point of intersection as the shared time and depth; the extension has them graph the same system.

All materials for 8.EE.C.8 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-23.

Objectives

Students will write two linear equations that model a draining tank and a filling tank.
Students will solve the system to find the time and depth where the two tanks match.

Vocabulary

Key words: system of equations, intersection, solution

Teacher overview

At a pump station, one tank often drains while another fills, and operators want to know the moment their depths match. That moment is the solution of a system of two linear equations. This set uses the fictional Hartsell pump station, where Tank A drains and Tank B fills at steady rates. Students build a table, read the crossing point from it, and then confirm the answer with algebra. A common error is stopping after finding the time and forgetting the depth; the solution of a system is both numbers, the time and the shared depth where the two lines cross.

Hartsell Station Two Tanks: A Systems of Equations Problem Set — Grade 8 Math

Name: ____________________________ Date: ______________

Problems

At the fictional Hartsell station, Tank A starts 20 feet deep and drains 0.4 feet each minute. Tank B starts 5 feet deep and fills 0.6 feet each minute. Let t stand for the number of minutes.

Write an equation for Tank A's depth A after t minutes.
Write an equation for Tank B's depth B after t minutes.
Make a table of each tank's depth at t = 0, 5, 10, and 15 minutes.
Using your table, at about what time do the two depths look equal?
Solve the system algebraically. Set 20 - 0.4t = 5 + 0.6t and find t.
What is the equal depth at that time? Use your t in either equation.
Before the tanks are equal, which one is deeper? After they are equal, which one is deeper?
Tank A must not drop below 8 feet. At what time does Tank A reach 8 feet?
Is the time in problem 8 before or after the tanks are equal?
Suppose Tank B had started at 8 feet instead of 5 feet, so B = 8 + 0.6t. Solve the new system 20 - 0.4t = 8 + 0.6t for t.

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

Answer Key

Answer: A = 20 - 0.4t. The start depth is 20 feet and it drops 0.4 feet per minute.
Answer: B = 5 + 0.6t. The start depth is 5 feet and it rises 0.6 feet per minute.
Answer: Tank A reads 20, 18, 16, 14 feet; Tank B reads 5, 8, 11, 14 feet. Each Tank A value drops by 2 every 5 minutes, and each Tank B value rises by 3 every 5 minutes.
Answer: 15 minutes. At t = 15 both tanks read 14 feet in the table.
Answer: 15 minutes. 20 - 0.4t = 5 + 0.6t. Add 0.4t to both sides: 20 = 5 + t. Subtract 5: 15 = t.
Answer: 14 feet. Use t = 15: Tank B is 5 + 0.6(15) = 5 + 9 = 14 feet, and Tank A is 20 - 0.4(15) = 20 - 6 = 14 feet.
Answer: Tank A is deeper before, Tank B is deeper after. Tank A keeps dropping while Tank B keeps rising, so they trade places at 15 minutes.
Answer: 30 minutes. 20 - 0.4t = 8, so -0.4t = -12, and t = 30.
Answer: After. Tank A reaches 8 feet at 30 minutes, which is later than the 15-minute crossing point.
Answer: 12 minutes. 20 - 0.4t = 8 + 0.6t. Add 0.4t: 20 = 8 + t. Subtract 8: 12 = t. Check: 20 - 4.8 = 15.2 and 8 + 7.2 = 15.2 feet.

Differentiation

Support

Extend the table in problem 3 to t = 20 and t = 25 so students can see the crossing point clearly before solving with algebra. Provide the first algebra step ("add 0.4t to both sides") already written for problem 5.

Extension

Ask students to graph both equations on the same grid, mark the intersection, and explain how the graph, the table, and the algebra all give the same solution. Then have them predict how the crossing time changes if Tank A drains faster.

ELL supports

Pre-teach "drain," "fill," and "intersection" with a two-tank sketch showing one arrow down and one arrow up. Provide a sentence frame for problem 7: "Before 15 minutes, Tank ___ is deeper, because ___."