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Activities for Intermediate Algebra: Instructors’ Guide

Katherine Yoshiwara

Section 2.3 Algebraic Solution of Systems

This Lesson is a review of material from elementary algebra: solving systems by substitution and elimination. We try to avoid artificial applications such as coin and age problems, considering instead weighted averages, which include mixture and interest problems. Finding the equilibrium price for supply and demand equations, and the break-even point from cost and revenue are also included.

Activity 2.3.1. Interest Problems.

Carmella has $1200 invested in two stocks; one earns 8% per year and the other earns 12% per year. The income from the 8% stock is $3 more than the income from the 12% stock. How much did she invest in each stock?
  1. Choose variables for the two unknown quantities, and fill in the table.
    \(\hphantom{0000} x\text{:}\)
    \(\hphantom{0000} y\text{:}\)
    Principal Interest Rate Interest
    First Stock \(\hphantom{0000000000}\) \(\hphantom{00000}\) \(\hphantom{00000}\)
    Second Stock
    Total
  2. Write one equation about the amount Carmella invested.
  3. Write a second equation about Carmella’s annual interest.
  4. Solve the system and answer the question in the problem.

Activity 2.3.2. Mixture Problems.

Amal plans to make 10 liters of a 17% acid solution by mixing a 20% acid solution with a 15% acid solution. How much of each should she use?
  1. Choose variables for the two unknown quantities, and fill in the table.
    \(\hphantom{0000} x\text{:}\)
    \(\hphantom{0000} y\text{:}\)
    Liters % Acid Amount of Acid
    First Solution \(\hphantom{0000000000}\) \(\hphantom{00000}\) \(\hphantom{00000}\)
    Second Solution
    Mixture
  2. Write one equation about the amount of solution Amal needs.
  3. Write a second equation about the amount of acid in the solutions.
  4. Solve the system and answer the question in the problem.

Activity 2.3.3. Motion Problems.

Because of prevailing winds a flight from Detroit to Denver, a distance of 1120 miles, takes 4 hours on Econoflite, while the return trip takes 3.5 hours. What were the speed of the airplane and the speed of the wind?
  1. Choose variables for the two unknown quantities, and fill in the table.
    \(\hphantom{0000} x\text{:}\)
    \(\hphantom{0000} y\text{:}\)
    Rate Time Distance
    Detroit to Denver \(\hphantom{0000000000}\) \(\hphantom{00000}\) \(\hphantom{00000}\)
    Denver to Detroit
  2. Write one equation about the trip from Detroit to Denver.
  3. Write a second equation about the return trip.
  4. Solve the system and answer the question in the problem.

Subsection 2.3.1 Check Your Understanding

  1. For the interest problem in Activity 1, we wrote one equation about and a second equation about .
  2. For the mixture problem in Activity 2, we wrote one equation about and a second equation about .
  3. If you are traveling upstream in a river, how does the current affect your speed?
  4. Which of the three types of problems, interest, mixture, or motion, is the easiest?

Subsection 2.3.2 Wrap Up

In this Lesson, we worked on the following skills and goals related to linear models:
  • Solve systems using substitution.
  • Solve systems using elimination.
  • Model and solve applied problems with a system of equations.

Subsection 2.3.3 Questions for Writing or Discussion

  1. When is the substitution method easier to use than the elimination method. Why?
  2. Explain how to use linear combinations to identify inconsistent and dependent systems.
  3. What is a weighted average?
  4. Explain why the solution of a system is also a solution of any linear combination of the two equations in the system.

Concept Questions.

  1. If you mix equal amounts of a 30% solution of insecticide and a 60% solution, what will be the strength of the mixture?
    1. 90%
    2. 45%
    3. 50%
    4. Need more information
  2. If you mix 2 cups of a 30% solution of insecticide with 3 cups of a 60% solution, what will be the strength of the mixture?
    1. 45%
    2. More than 45%
    3. Less than 45%
    4. Need more information
  3. The chemistry department has 80 students, of whom 35% are women. The physics department has 60 students, of whom 15% are women. What is the percent of women in both departments?
    1. 25%
    2. More than 25%
    3. Less than 25%
    4. Need more information
  4. Delbert has 4 test grades of 50%, and then got 100% on the fifth test. What is his test average now?
    1. 75%
    2. 60%
    3. 90%
    4. 55%
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