LESSON 3 • Systems of Linear Equations 57

Summarize the Mathematics

In this investigation, you examined another method for solving a system of linear equations—the elimination method.a What would be your first two steps in solving the system

2x + y = -9 ⎧ ⎨

⎩

4x + 3y = 1 using the

elimination method? Justify each step.

b You now have worked with three quite different methods for solving systems of linear equations. How do you decide on a method to use in any particular problem?

c Is it possible to use any of the three methods when solving any system of two linear equations?

Be prepared to share your ideas and reasoning with the class.

Check Your UnderstandingCheck Your UnderstandingAs you complete these tasks, think about the advantages and disadvantages of using the elimination method for solving a system of linear equations.

a. Use the elimination method to solve each of these systems of linear equations. Check your solutions.

i. 2x + y = 6⎧ ⎨

⎩

3x + 5y = 16

ii. 6x - y = 18⎧ ⎨

⎩

2x + y = 2

iii. 2x + 3y = 5⎧ ⎨

⎩

-3x + 2y = -14

b. Given a choice between graphing, substitution, and elimination, which would you choose to solve each of the systems in Part a? Explain the reasoning for your choices.

IInnvest ivest iggationation 33 Systems with Zero and Infinitely Systems with Zero and Infinitely Many SolutionsMany Solutions

Systems of linear equations do not always have solutions consisting of a single ordered pair (x, y). As you work through this investigation, look for answers to this question:

What are the properties of linear systems that do not have exactly one ordered pair solution?

Name:________________________________Unit 1Lesson 3

Investigation 3

58 UNIT 1 • Functions, Equations, and Systems

1 Think about the graphing method for estimating solutions to a system of two linear equations.

a. Suppose (5, 8) is the solution to a system of linear equations. How could you see this by looking at the graphs of the equations?

b. Sketch graphs of a system of linear equations that has no solution.

c. Is it possible for a system of linear equations to have infinitely many solutions? What would the graphs look like in this case?

2 For some people, like athletes and astronauts, selection of a good diet is a carefully planned scientific process. In the case of astronauts, proper nutrition is provided in limited forms. For example, drinks might come in disposable boxes and solid food in energy bars. Suppose that, in planning daily diets for a space shuttle team, nutritionists work toward these goals.

• Drinks each provide 30 grams of carbohydrate, energy bars each provide 40 grams of carbohydrate, and the optimal diet should contain 600 grams of carbohydrate per day.

• Drinks each provide 15 grams of protein, energy bars each provide 20 grams of protein, and the optimal diet should contain 200 grams of protein.

The problem is to find a number of drinks and a number of energy bars that will provide just the right nutrition for each astronaut. If we use x to represent the number of drinks and y for the number of energy bars, the goals in diet planning can be expressed as a system of linear equations:

30x + 40y = 600 ⎧ ⎨

⎩

15x + 20y = 200

a. What does the first equation represent? The second?

Solving that system by the elimination method might follow steps like these.

Step 1: 30x + 40y = 600⎧ ⎨

⎩

30x + 40y = 400

Step 2: 0 = 200

LESSON 3 • Systems of Linear Equations 59

b. What properties of equations justify Steps 1 and 2 in this solution method?

c. What does the equation in Step 2 say about the possibility of finding (x, y) values that satisfy the original system?

d. The diagram at the right shows graphs of solutions to the two original linear equations. How does it help explain the difficulty in finding a solution for the system?

3 Suppose that the condition on protein in Problem 2 was revised to require 300 grams per day.

a. Write the new system of equations expressing the conditions relating number of drink boxes and number of food bars to the required grams of carbohydrate and protein in the diet.

b. Use the elimination method to solve this new system. What happens? What does this tell you?

c. Sketch a graph for each equation in the system in Part a. What do the graphs tell you about solutions for this system?

4 Solve each of the following systems of linear equations in two ways:

• first by estimation using graphs of the two equations,

• then by either substitution or elimination.

In each case, be prepared to explain what the result tells about solutions for the system and how that result is demonstrated in the graph and in the algebraic solution process.

a. x - y = 4⎧

⎨ ⎩

2x - 2y = 8 b.

3x - 6y = -46.5 ⎧ ⎨

⎩

-x + 2y = 15.5

c. y = -5x + 12⎧

⎨ ⎩

y = -5x - 7 d.

x - 2y = 0 ⎧ ⎨

⎩

3x - 5y = 2.5

5 For which of the systems in Problem 4 could you determine the number of solutions just by examining the equations? How could you tell?

6 In the problems of this lesson, you’ve studied systems of linear equations that have unique solutions—a single pair of values for the variables that satisfy both equations—and others that have infinitely many solutions or no solutions at all. Find values of a, b, and c that

will guarantee the system 3x - 2y = 4 ⎧ ⎨

⎩

ax + by = c has:

a. exactly one solution (x, y).

b. infinitely many solutions.

c. no solutions.

00

5 10 15 20 25

5

10

15

x

y