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Worksheet Grade 8 Math

Ferngap Dam Release Schedule: Solving Linear Equations

A grade 8 worksheet on writing and solving one-variable linear equations for two reservoir levels, including equations with variables on both sides.

Objectives

  1. Students will write a linear equation for a reservoir level that changes at a steady rate.
  2. Students will solve linear equations, including equations with variables on both sides.

Vocabulary

Key words: linear equation, variable, solution

Teacher overview

A dam operator plans releases days in advance, and a steady rise or fall in the water level is exactly a linear relationship. This worksheet uses a fictional dam, Ferngap Dam, and its two pools to give students equations worth solving: some have the variable on one side, some on both sides, and two of them have no solution or infinitely many solutions. Students see that the same algebra answers a real scheduling question. A common error is subtracting a term from only one side of the equation; remind students that whatever they do to one side, they must do to the other.

Ferngap Dam Release Schedule: Solving Linear Equations — Grade 8 Math
Name: ____________________________ Date: ______________

Student Worksheet

Ferngap Dam holds back a lower reservoir and a smaller upper pool. Both are fictional. Each level is the water depth read on that pool's own gauge. During a dry stretch with almost no inflow, the lower reservoir starts at a level of 150 feet and drops 2 feet each day. Let t stand for the number of days into the dry stretch.

  1. Write an equation for the lower reservoir's level L after t days.
  2. Operators must keep the level at or above 118 feet. Solve 150 - 2t = 118 to find the day it reaches exactly 118 feet.
  3. A boat ramp closes when the level falls below 104 feet. On what day does the level reach 104 feet?
  4. The upper pool starts at 90 feet and rises 1 foot each day as it fills. Write an equation for its level U after t days.
  5. On what day are the two pools at the same level? Solve 150 - 2t = 90 + t.
  6. What is that equal level, in feet? Use your day from item 5 in either equation.
  7. A different fill plan starts the upper pool at 96 feet. Solve 150 - 2t = 96 + t for t.
  8. A report claimed the lower reservoir followed both L = 150 - 2t and L = 158 - 2t on the same days. Solve 150 - 2t = 158 - 2t. What does your result tell you?
  9. A note said the level followed both 150 - 2t and 2(75 - t). Solve 150 - 2t = 2(75 - t). What does your result tell you?
  10. In item 5 the two pools meet on day 20. Explain in one sentence which pool is higher before day 20 and which is higher after it.

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: L = 150 - 2t. The start value is 150 feet and the level drops 2 feet per day, so the day count t is multiplied by -2.
  2. Answer: 16 days. 150 - 2t = 118, so -2t = -32, and t = 16.
  3. Answer: 23 days. 150 - 2t = 104, so -2t = -46, and t = 23.
  4. Answer: U = 90 + t. The start value is 90 feet and the level rises 1 foot per day, so t is added.
  5. Answer: 20 days. 150 - 2t = 90 + t. Add 2t to both sides: 150 = 90 + 3t. Subtract 90: 60 = 3t, so t = 20.
  6. Answer: 110 feet. Use t = 20: 90 + 20 = 110 feet, and the lower reservoir gives 150 - 2(20) = 110 feet as a check.
  7. Answer: 18 days. 150 - 2t = 96 + t. Add 2t: 150 = 96 + 3t. Subtract 96: 54 = 3t, so t = 18. Check: 150 - 36 = 114 and 96 + 18 = 114.
  8. Answer: No solution. Adding 2t to both sides leaves 150 = 158, which is false, so no day makes both descriptions true; the two claims cannot both be correct.
  9. Answer: Infinitely many solutions. The right side 2(75 - t) equals 150 - 2t, so the equation becomes 150 - 2t = 150 - 2t, which is true for every t; the two forms describe the same line.
  10. Answer: The lower reservoir is higher before day 20. After day 20 the upper pool is higher, because the lower level keeps dropping while the upper level keeps rising.

Differentiation

Support

Provide a two-column table for items 2, 3, 5, and 7 so students can substitute a few day values and see the level approach the target before they solve with algebra. Give a worked model for a variables-on-both-sides equation using different numbers before students try item 5.

Extension

Ask students to find the day when the upper pool is exactly 10 feet higher than the lower reservoir, writing and solving the equation 90 + t = (150 - 2t) + 10. Have them explain why the answer is later than day 20.

ELL supports

Pre-teach "reservoir," "level," "rise," and "drop" with a simple side-view sketch of a dam and two labeled pools. Provide sentence frames for items 8, 9, and 10: "There is no day because ___," "Every day works because ___," and "Before day 20 the ___ pool is higher."