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Worksheet Grade 8 Math

Ryegate Treatment Plant: Solving Linear Equations for Daily Operations

A grade 8 worksheet of one-variable linear equations at a fictional treatment plant, with variables on both sides and one, none, or infinite solutions.

Objectives

  1. Students will solve one-variable linear equations, including equations with variables on both sides.
  2. Students will recognize equations that have one solution, no solution, or infinitely many solutions.

Vocabulary

Key words: variable, solution, distributive property

Teacher overview

Plant operators plan a shift by setting a target and solving for the unknown that reaches it, which is exactly what a one-variable equation does. This worksheet turns that daily planning into linear-equation practice, including three equations with variables on both sides and two special cases: one with no solution and one with infinitely many. A common mistake to watch for is stopping too early on the special cases; when the variables cancel, students should read what is left (a false statement means no solution, a true statement means every value works) instead of writing "x = 0."

Ryegate Treatment Plant: Solving Linear Equations for Daily Operations — Grade 8 Math
Name: ____________________________ Date: ______________

Student Worksheet

Ryegate Water Treatment Plant is a fictional plant. Its operators use equations to plan each shift. Solve each equation for the unknown, and show your steps. For items 6 and 7, state whether there is one solution, no solution, or infinitely many solutions.

  1. A backwash cycle uses 250 gallons. After c cycles the plant has used 250c gallons. One day the plant used 3,500 gallons of backwash. Write and solve 250c = 3,500 to find the number of cycles.
  2. The clearwell starts 8 feet deep and rises 0.4 foot each hour, so its depth is d = 0.4t + 8 after t hours. Solve 0.4t + 8 = 14 to find when it reaches 14 feet.
  3. Solve for x: 5x + 30 = 8x.
  4. Solve for x: 6(x - 2) = 42.
  5. Solve for x: 4(x + 5) = 2x + 26.
  6. An operator writes 3x + 7 = 3x - 2. Solve for x, and state how many solutions the equation has.
  7. Solve 2(4x + 6) = 8x + 12, and state how many solutions the equation has.
  8. A dosing plan follows the equation 0.5x + 1.2 = 3.7, where x is the number of doses. Solve for x.
  9. On a busy day the plant treats 900,000 gallons, which is 250,000 gallons more than twice a slow day's amount s. Write and solve 2s + 250,000 = 900,000.
  10. Solve for x: 5x - 3 = 3x + 11.

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: 14 cycles.
    250c = 3,500, so c = 3,500 / 250 = 14 cycles.
  2. Answer: 15 hours.
    0.4t + 8 = 14, so 0.4t = 6 and t = 6 / 0.4 = 15 hours.
  3. Answer: x = 10.
    5x + 30 = 8x. Subtract 5x from both sides: 30 = 3x, so x = 10.
  4. Answer: x = 9.
    6(x - 2) = 42. Divide both sides by 6: x - 2 = 7, so x = 9.
  5. Answer: x = 3.
    4(x + 5) = 2x + 26 becomes 4x + 20 = 2x + 26. Subtract 2x: 2x + 20 = 26. Subtract 20: 2x = 6, so x = 3.
  6. Answer: no solution.
    3x + 7 = 3x - 2. Subtract 3x from both sides to get 7 = -2, which is false, so no value of x works.
  7. Answer: infinitely many solutions.
    2(4x + 6) = 8x + 12 becomes 8x + 12 = 8x + 12. The two sides are identical, so every value of x works.
  8. Answer: x = 5.
    0.5x + 1.2 = 3.7, so 0.5x = 2.5 and x = 2.5 / 0.5 = 5 doses.
  9. Answer: 325,000 gallons.
    2s + 250,000 = 900,000, so 2s = 650,000 and s = 325,000 gallons.
  10. Answer: x = 7.
    5x - 3 = 3x + 11. Subtract 3x: 2x - 3 = 11. Add 3: 2x = 14, so x = 7.

Differentiation

Support

Provide a two-column "steps and reasons" organizer for items 3, 5, and 10 so students record each move (subtract, add, divide) beside each line. Work one variables-on-both-sides example together before students start, and allow a calculator for the decimal steps in items 2 and 8.

Extension

Ask students to write their own operations equation with variables on both sides that has exactly one solution, then change one number so it has no solution, and a second number so it has infinitely many. They should show why each version behaves the way it does.

ELL supports

Pre-teach "solve," "solution," and "both sides" with a balance-scale picture that shows keeping the equation even. Provide the sentence frames "There is ___ solution because ___" for items 6 and 7.