lesson 17: solving systems of equations - mr...

Hart Interactive – Algebra 1

M2 Lesson 17 ALGEBRA I

Lesson 17: Solving Systems of Equations Opening Exercise

In this lesson, we’ll look at various word problems and use substitution or graphing to solve them. Let’s look at a problem from Lewis Carroll’s Through the Looking Glass when Alice encounters Tweedledum and Tweedledee.

1. Tweedledum says to Tweedledee, “The sum of your weight and twice mine is 361 pounds.” Tweedledee replies, “The sum of your weight and twice mine is 362 pounds.” Help Alice find both of their weights.

[source: John Tenniel's illustration, from Through the Looking-Glass (1871), chapter 4]

A. First define your variables: Let x = __________________ weight and y = __________________ weight.

B. Write an equation for each statement. ____________________

____________________

C. Solve the system of equations using substitution. You’ll need to get one variable isolated in one of the equations.

D. Check that your answer makes sense.

The sum of your weight and twice

mine is 362 pounds.

The sum of your weight and twice

mine is 361 pounds.

Lesson 17: Solving Systems of Equations Unit 4: Systems of Equations and Inequalities

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2. A. Graph the two equations from the comic strip on

the grid provided.

B. Where do you see Paige’s answer of shirts are $15 and sweaters are $30 on the graph?

C. Why was Paige able to solve the problem her brother gave her but she couldn’t solve the problem on her worksheet?

3. Discuss with your partner how you could have solved Paige’s problem using substitution.


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4. Lulu tells her little brother, Jack, that she is holding 20 coins, all of which are either dimes or quarters. They have a value of $4.10. She says she will give him the coins if he can tell her how many of each she is holding. Solve this problem for Jack.

A. First define your variables: Let D = __________________ and Q = __________________.

B. Write an equation for each statement. ____________________

____________________

C. Solve the system of equations by substitution.

D. Check that your answer makes sense.

5. Tickets to a school play cost $5 for adults and $3 for children. One day they sold 175

tickets and brought in $675. How many children tickets were sold that day?


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6. At a state fair, there is a game where you throw a ball at a pyramid of cans. If you knock over all of the cans, you win a prize. The cost is 3 throws for $1, but if you have an armband, you get 6 throws for $1. The armband costs $10.

A. CHALLENGE Write two cost equations for the game in terms of the number of

throws purchased, one without an armband and one with. Let t = number of throws Let c = total cost

B. Graph the two cost equations on the same graph. Think about whether this data is discrete or continuous.

C. Does it make sense to buy the armband? Explain your thinking.

[image source: texasentertainmentgroup.com]

02468

1012141618202224262830

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90

Cost

in D

olla

rs

Number of Throws

Can Smash Game


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Identifying the Correct System

7. For the situation below, determine which system of equations fits the scenario. (You do not have to solve the problem.)

At Elisa’s Printing Company there are two kinds of printing presses. Model A

can print 70 books per day and Model B can print 55 books per day. The

company owns 14 total printing presses and this allows them to print 905 books

per day. How many of each type of press do they have?

Let x = the number of presses that can print 70 books per day

Let y = the number of presses that can print 55 books per day

A. 905

70 55 14x y

x y+ =+ =

B. 905

70 55 905x y

x y+ =+ =

C. 14

70 55 905x y

x y+ =+ =

8. For the situation below, determine which system of equations fits the scenario. (You do not have to solve the problem.)

Elise is four years older than Megan. If the sum of their ages is 16, how old are

Elise and Megan?

Let e = the age of Elise

Let m = the age of Megan

A. 16

4e me m+ =+ =

B. 16

4e me m+ == +

C. 164

e me m+ =+ =


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9. A. What are the advantages of finding the solution by graphing? B. What are the advantages of finding the solution algebraically?

C. If a system of linear equations had the same slope and the same y-intercept, what would the solution be?

D. If a system of linear equations had the same slope and different y-intercepts, what would the solution be?


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Lesson Summary

When writing systems of equations, be sure to define your variables, write your equations, solve your system of equations and then check if your answer makes sense.

Example: Adriana is thinking of two numbers. She says that one number is one more than twice the other number and the difference of the numbers is 7. What are Adriana’s numbers?

Let x = one of the numbers and y = the other number.

�𝑦𝑦 = 2𝑥𝑥 + 1𝑦𝑦 − 𝑥𝑥 = 7

Solve the system of equations.

Graphically:

Algebraically:

The solution to the system of equations is ______________.

10. Finish the Lesson Summary problem and find the solution to the problem.

11. CHALLENGE If the second equation was written as x – y = 7, what would be the solution? Does this work for Adriana’s numbers?


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Homework Problem Set

Solve each system first by graphing and then algebraically.

1. �𝑦𝑦 = 4𝑥𝑥 − 1𝑦𝑦 = −1

2 𝑥𝑥 + 8 Solution: ___________

2. � 2𝑥𝑥 + 𝑦𝑦 = 42𝑥𝑥 + 3𝑦𝑦 = 9 Solution: ___________

3. �3𝑥𝑥 + 𝑦𝑦 = 53𝑥𝑥 + 𝑦𝑦 = 8 Solution: ___________


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4. For each question below, provide an explanation or an example to support your claim. A. Is it possible to have a system of equations that has no solution? B. Is it possible to have a system of equations that has more than one solution?

5. Without graphing, construct a system of two linear equations where (0, 5) is a solution to the first equation but is not a solution to the second equation, and (3, 8) is a solution to the system.

6. Consider two linear equations. The graph of the first equation is shown. A table of values satisfying the second equation is given. What is the solution to the system of the two equations?

𝒙𝒙 −4 −2 0 2 4 𝒚𝒚 −26 −18 −10 −2 6


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7. Pam has two part time jobs. At one job, she works as a cashier and makes $8 per hour. At the second job, she works as a tutor and makes $12 per hour. One week she worked 30 hours and made $268. How many hours did she spend at each job? Be sure to show you work and explain your thinking.


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lesson 17: solving systems of equations - mr...

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