Math problem set Grade 8 Science

Saltmarsh Cove Water Blending: A System of Equations

Grade 8 students write and solve a system of two equations to blend reverse-osmosis water with brackish well water to a target at a fictional coastal plant.

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

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: Problems 1 through 3 and problem 9 set up and solve a system of two linear equations, a total-flow equation and a dissolved-solids balance, to find the blend, the real-world system this standard targets.

All materials for 8.EE.C.8 →

MS-ESS3-1 NGSS

Construct a scientific explanation based on evidence for how the uneven distributions of Earth's mineral, energy, and groundwater resources are the result of past and current geoscience processes.

How this material aligns: The scenario treats the brackish well as a locally available but salty groundwater resource that must be desalinated or blended, connecting the math to the uneven groundwater distribution this standard explains.

All materials for MS-ESS3-1 →

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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 a system of two linear equations for a water-blending problem.
Students will solve the system and interpret the solution as flows and percentages.

Vocabulary

Reverse osmosis: Reverse osmosis cleans water by pushing it through a membrane with holes so tiny that salt and most other dissolved things cannot fit through. It can even turn seawater into fresh water.

Also: system of equations, brackish, dissolved solids

Teacher overview

Some coastal towns get fresh water by pushing brackish (slightly salty) groundwater through reverse-osmosis membranes, but running every gallon through the membranes uses a lot of energy. Blending very clean membrane water with a smaller amount of salty well water can hit a target while saving power. This set uses the fictional town of Saltmarsh Cove to build that blend with a system of equations. A common misconception is to simply average the two dissolved-solids numbers, 50 and 1,550 milligrams per liter, and expect 800 to be the blend; the real blend depends on how much of each source is used, which is why a dissolved-solids balance is needed.

Saltmarsh Cove Water Blending: A System of Equations — Grade 8 Science

Name: ____________________________ Date: ______________

Problems

Saltmarsh Cove (a fictional coastal town) runs a reverse-osmosis plant that treats water from its brackish well. The membrane water, called permeate, is very clean at 50 mg/L of total dissolved solids (TDS). The untreated brackish well water is 1,550 mg/L TDS. The plant wants 10 MGD (million gallons per day) of finished water at a target of 500 mg/L TDS. Let x be the MGD of permeate and y be the MGD of brackish well water.

Write an equation showing that the permeate flow and the brackish flow add up to the 10 MGD of finished water.
Write the dissolved-solids balance: 50x plus 1,550y must equal the target 500 mg/L times 10 MGD.
Solve the system from problems 1 and 2. How many MGD of permeate and how many MGD of brackish water are blended?
What percent of the finished water comes from the brackish well?
Making permeate from this brackish water uses about 4 kWh per 1,000 gallons. Using your permeate flow from problem 3, how many kWh per day does it take to make that permeate? (Remember 1 MGD is 1,000,000 gallons per day.)
If the town skipped blending and made all 10 MGD by reverse osmosis at 4 kWh per 1,000 gallons, how many kWh per day would that take?
How many kWh per day does blending save compared with treating all the water by reverse osmosis? Subtract your problem 5 answer from your problem 6 answer.
The brackish well by itself is 1,550 mg/L. Explain why the town cannot simply deliver the well water without blending or treatment, using the 500 mg/L target.
Suppose the target tightens to 350 mg/L, with the total still 10 MGD and the same two sources. Set up and solve the new system for x and y.
Compare your answers from problems 3 and 9. Explain what happens to the amount of brackish water in the blend as the target TDS gets lower.

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: x + y = 10.
The two flows must add to the 10 MGD of finished water.
Answer: 50x + 1,550y = 5,000.
The target load is 500 mg/L times 10 MGD, which is 5,000, so 50x + 1,550y = 5,000.
Answer: x = 7 MGD of permeate and y = 3 MGD of brackish water.
From x + y = 10, x = 10 − y. Substitute: 50(10 − y) + 1,550y = 5,000, so 500 + 1,500y = 5,000, then 1,500y = 4,500 and y = 3. Then x = 7. Check: 50(7) + 1,550(3) = 350 + 4,650 = 5,000.
Answer: 30 percent.
3 MGD of brackish out of 10 MGD total is 3 ÷ 10 = 0.30, or 30 percent.
Answer: 28,000 kWh per day.
7 MGD is 7,000,000 gallons. That is 7,000 units of 1,000 gallons, and 7,000 × 4 = 28,000 kWh per day.
Answer: 40,000 kWh per day.
10 MGD is 10,000,000 gallons, which is 10,000 units of 1,000 gallons, and 10,000 × 4 = 40,000 kWh per day.
Answer: 12,000 kWh per day.
40,000 − 28,000 = 12,000 kWh per day saved by blending.
Answer: The well is far above the target, so it must be diluted.
At 1,550 mg/L the well water is more than three times the 500 mg/L target, so delivering it alone would fail the target; blending with clean permeate brings the dissolved solids down. Accept any answer that compares 1,550 to 500.
Answer: x = 8 MGD of permeate and y = 2 MGD of brackish water.
New balance: 50x + 1,550y = 350 × 10 = 3,500, with x + y = 10. Substitute x = 10 − y: 500 + 1,500y = 3,500, so 1,500y = 3,000 and y = 2. Then x = 8. Check: 50(8) + 1,550(2) = 400 + 3,100 = 3,500.
Answer: Less brackish water is used as the target gets lower.
The brackish share dropped from 3 MGD to 2 MGD when the target fell from 500 to 350 mg/L, because a cleaner target leaves less room for high-TDS well water. Accept any answer describing that a lower target means less brackish water.

Differentiation

Support

Give students the two equations already written for problems 1 and 2 so they can focus on solving the system by substitution. Provide a worked example with different numbers (a target that gives a whole-number answer) before problem 3, and allow a calculator throughout.

Extension

Ask students to find the target TDS that would require an equal blend of 5 MGD permeate and 5 MGD brackish water, then explain how they used the dissolved-solids balance to find it. Have them describe how the energy savings change as more brackish water is blended in.

ELL supports

Pre-teach "permeate," "brackish," and "dissolved solids" with a simple diagram of salty water going through a membrane and coming out clean. Provide the frame "___ plus ___ equals 10" for problem 1. Allow students to show the substitution steps rather than explaining them in sentences.