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Unit 1 > Functions, Systems, and Equations

Lesson 3: Systems of Linear Equations Practice Problems

I can write and solve systems of linear equations through a variety of methods. !

Investigation Practice Problem

Options Max Possible

Points Total Points

Earned Investigation 1: Solving with Graphs

and Substitution #1, 2, 3 12 points

Investigation 2: Solving by Elimination #4, 5 6 points

Investigation 3: Systems with Zero and Infinitely Many Solutions

#6, 7, 8, 9, 10

18 points

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________/36 points !

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**Check your work with Mrs. Yauk’s answer key once you have earned your points for these practice problems, and earn 5 extra credit points – you must use a different color and write in

the correct answer for any problems you get incorrect**

LESSON 3 • Systems of Linear Equations 61

On Your Own

Applications

1 To participate in a school trip, Kim had to earn $85 in one week. Kim could earn $8 per hour babysitting and $15 per hour for yard work, but Kim’s parents limit work time to 8 hours per week.

a. Write two equations, one that represents the condition on total number of hours to be worked and the other which relates the number of hours worked at each job toward the fund-raising goal.

b. Solve the system of equations you wrote in Part a to find out how many hours Kim will have to work at each job to exactly meet the income goal and the time constraint.

2 Solve the following systems of equations, using graphing as the method for one of them.

a. y = 3x - 5⎧

⎨ ⎩

8x - 4y = 30 b.

y = 5x ⎧ ⎨

⎩

6x - 2y = 12 c.

y = 5x + 15⎧ ⎨

⎩

y = -x - 3

3 The relationship between supply and demand is important in the business world. A supply function indicates the relationship between price of a product and the amount of that product that will be available from suppliers. A demand function indicates the relationship between price of a product and the amount of the product that will be purchased by consumers. An electronics store devised the following functions to study supply and demand for its best-selling DVD players.

Demand: y = -0.5x + 90, where y stands for the estimated number sold, and x stands for the price of the DVD player.

Supply: y = 1.5x - 30, where y stands for the number available, and x represents the price of the DVD player.

a. Why do you suspect the demand function has a negative slope but the supply function’s slope is positive?

b. Equilibrium is reached when supply is equal to demand. At what price should the store sell the DVD player in order to reach equilibrium between supply and demand?

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62 UNIT 1 • Functions, Equations, and Systems

On Your Own

4 Carly is training for an upcoming fitness competition and is trying to find a breakfast combination that meets her nutritional requirements of 950 calories and 25 grams of protein. One serving of her cereal of choice has 200 calories and 2 grams of protein. Her favorite brand of peanut butter contains 180 calories and 8 grams of protein per serving.

a. Write an equation that relates the number of servings of cereal and the number of servings of peanut butter to the total number of calories she needs for breakfast.

b. Write another equation that relates the number of servings of cereal and the number of servings of peanut butter to the total amount of protein she needs for breakfast.

c. What numbers of servings for each type of food would meet both of her nutrition goals?

5 Laura and Andy are trying to earn money to buy airplane tickets to visit their favorite aunt, Annie. Laura’s ticket is going to cost her $280 while Andy found a ticket for $230 on the Internet. To earn their money, they have both decided to mow lawns and babysit. Laura charges $7 per hour for babysitting while Andy charges $5 per hour. To mow a lawn, Laura charges $14 per lawn while Andy charges $16 per lawn.

a. Write an equation relating income from Laura’s work to her ticket cost. Use B to represent number of hours babysitting and L to represent number of lawns mowed. Use the same variables to write another equation relating income from Andy’s work to his ticket cost.

b. Is it possible that Laura and Andy could each reach their ticket price goal by mowing the same number of lawns and babysitting the same number of hours? If so, find those numbers; if not, explain how you know.

6 Solve the following systems of equations and check your solutions. Show that you know how to use different solution strategies by using each of the three methods once—graphing, substitution, and elimination. Be prepared to explain your choice of solution method in each case.

a. -2x - y = 3 ⎧ ⎨

⎩

x + 2y = 4 b.

x = 5 - 2y⎧ ⎨

⎩

3x - y = 15 c.

-2x + y = 3⎧ ⎨

⎩

-4x + 2y = 2

7 For each system below, without graphing, determine if the lines represented by the equations are the same, are different and intersect at a point, or are different and parallel. Explain your reasoning in each case. If there is exactly one solution, find and check it.

a. y = -4x + 5 ⎧ ⎨

⎩

y = -4x - 2 b.

y = 6x - 2⎧ ⎨

⎩

y = 3x - 2 c.

y = 1.5x + 9⎧ ⎨

⎩

y = 3 _ 2 x + 9

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LESSON 3 • Systems of Linear Equations 63

On Your Own

8 Without graphing the equations, determine if the lines represented in each system below are the same, are different and intersect at a point, or are different and parallel. Explain your reasoning in each case. If there is exactly one solution, find and check it.

a. x + 2y = 8⎧

⎨ ⎩

2x + y = 4 b.

x - 3y = 6⎧ ⎨

⎩

3x - 9y = 18

c. 3x - 2y = 1⎧ ⎨

⎩

6x - 4y = 10 d.

x + 7 = 10⎧ ⎨

⎩

2x - 5y = 16

9 Find values of a, b, and c that will guarantee that the system

x + 2y = 3⎧ ⎨

⎩

ax + by = c has:

a. exactly one solution.

b. infinitely many solutions.

c. no solutions.

Connections

10 Imagine yourself as the manager of a local movie theater. Clearly, you are interested in both filling your movie theater and reaching a certain level of revenue.

a. Let x stand for the number of full-price tickets and y for the number of discounted tickets. Then write an equation that relates x and y to the theater capacity of 800 moviegoers.

b. To meet your goal of $6,000 ticket revenue, you have set ticket prices at $9 for full-price tickets and $4 for discounted tickets. Write another equation that relates the number of full-price tickets x and the number of discounted tickets y to the revenue goal of $6,000.

c. How many tickets of each type would you have to sell to fill the theater and meet your financial goal?

11 Suppose a system of two linear equations has the indicated number of solutions. In each case, what can you say about the slopes and y-intercepts of their graphs?

a. infinitely many solutions

b. no solutions

c. exactly one solution

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