Lesson 9 Can You Get to the Point, Too? Solidify Understanding

Learning Focus

Solve systems of linear equations by eliminating one of the variables.

How do I use the logical reasoning for solving the scenarios in the previous task when the scenarios are represented with linear equations in standard form?

Open Up the Math: Launch, Explore, Discuss

In Shopping for Cats and Dogs, Carlos found a way to find the cost of individual items when given the purchase price of two different combinations of those items. He would like to make his strategy more efficient by writing it out using symbols and algebra. Help him formalize his strategy by doing the following:

For each scenario in Shopping for Cats and Dogs write a system of equations to represent the two purchases.
Show how your strategies for finding the cost of individual items could be represented by manipulating the equations in the system. Write out intermediate steps symbolically, so that someone else could follow your work.
Once you find the price of one of the items in the combination, show how you would find the price of the other item.

1.

One week Carlos bought $3$ bags of Tabitha Tidbits and $4$ bags of Figaro Flakes for $$ 43.00$ . The next week he bought $3$ bags of Tabitha Tidbits and $6$ bags of Figaro Flakes for $$ 54.00$ . Based on this information, figure out the price of one bag of each type of cat food. Explain your reasoning.

2.

One week Carlos bought $2$ bags of Brutus Bites and $3$ bags of Milo Munchies for $$ 42.50$ . The next week he bought $5$ bags of Brutus Bites and $6$ bags of Milo Munchies for $$ 94.25$ . Based on this information, figure out the price of one bag of each type of dog food. Explain your reasoning.

3.

Carlos purchased $6$ dog leashes and $6$ cat brushes for $$ 45.00$ for Clarita to use while pampering the pets. Later in the summer he purchased $3$ additional dog leashes and $2$ cat brushes for $$ 19.00$ . Based on this information, figure out the price of each item. Explain your reasoning.

4.

One week Carlos purchased $2$ boxes of cat treats and $3$ boxes of dog treats for $$ 18.50$ . The next week, he bought $2$ box of cat treats and $2$ boxes of dog treats for $$ 14.00$ . The third week he bought $5$ boxes of both cat and dog treats for $$ 35.00$ . Based on this information, figure out the price of each item. Explain your reasoning.

5.

Carlos has noticed that because each of his purchases has been somewhat similar, it has been easy to figure out the cost of each item. However, his last set of receipts has him puzzled. One week he tried out cheaper brands of cat and dog food. On Monday he purchased $3$ small bags of cat food and $5$ small bags of dog food for $$ 22.75$ . Because he went through the small bags quite quickly, he had to return to the store on Thursday to buy $2$ more small bags of cat food and $3$ more small bags of dog food, which cost him $$ 14.25$ . Based on this information, figure out the price of each bag of the cheaper cat and dog food. Explain your reasoning.

Pause and Reflect

While working on each of these problems, at some point you not only eliminated a variable, but you probably also eliminated an equation. To remind ourselves that we are working with a system of equations, it might be helpful to write out the solution process as a sequence of equivalent systems. To do so, we can obtain an equivalent system of equations by replacing one or both equations in the system using one of the following actions:

Replace an equation in the system with a constant multiple of that equation.
Replace an equation in the system with the sum or difference of the two equations.
Replace an equation with the sum of that equation and a multiple of the other.

6.

Solve the following problem by keeping track of both equations in the system each step along the way. That is, each step in your solution process will be a system of two linear equations. Write an explanation that explains how you changed the previous system to get the next system. The final system in the sequence will be ${\begin{matrix} d = a number \\ c = a number \end{matrix}$

Carlos purchased $4$ dog ID tags and $12$ cat ID tags for $$ 36.00$ to keep track of the pet owner’s contact information. Later in the summer he purchased $10$ additional dog tags and $6$ cat tags for $$ 30.00$ . Based on this information, figure out the price of each item.

System	Explanation
${\begin{matrix} 4 d + 12 c = 36 \\ 10 d + 6 c = 30 \end{matrix}$	Original system for the constraints.
${\begin{matrix} 2 d + 6 c = 18 \\ 10 d + 6 c = 30 \end{matrix}$
${\begin{matrix} 8 d + 0 c = 12 \\ 10 d + 6 c = 30 \end{matrix}$

Ready for More?

Writing out each system of equations reminded Carlos of his work with solving systems of equations graphically. Show how the cost of each item shows up graphically in each system in the solution process for problem 6. Record each system, and then sketch its graph on a separate grid. You may need fewer or more grids than are given, depending upon the number of systems in your solution process.

a.

System: ${\begin{matrix} 4 d + 12 c = 36 \\ 10 d + 6 c = 30 \end{matrix}$

b.

System: ${\begin{matrix} 2 d + 6 c = 18 \\ 10 d + 6 c = 30 \end{matrix}$

c.

System: ${\begin{matrix} 8 d + 0 c = 12 \\ 10 d + 6 c = 30 \end{matrix}$

d.

System: ${\begin{matrix} d = 1.5 \\ 10 d + 6 c = 30 \end{matrix}$

e.

System: ${\begin{matrix} d = 1.5 \\ 6 c = 15 \end{matrix}$

f.

System: ${\begin{matrix} d = 1.5 \\ c = 2.5 \end{matrix}$

Takeaways

When solving equations, we write a sequence of equivalent equations until the solution to the equation is apparent.

Likewise, when solving systems of equations, we:

To create equivalent systems of equations we can:

Vocabulary

solve a system by elimination or substitution
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to solve systems of equations by eliminating one of the variables. To do so, we had to think of equations as objects that can be added, subtracted, or multiplied by a scale factor. Each operation creates an equivalent system of equations, and if we are strategic, we can get a system of equations for which the solutions to the system are apparent.

Retrieval

Determine the solution to each system of linear equations.

1.

$x$	$f (x)$
$- 3$	$7$
$- 2$	$4$
$- 1$	$1$
$0$	$- 2$
$1$	$- 5$

$x$	$g (x)$
$- 3$	$- 3$
$- 2$	$- 1$
$- 1$	$1$
$0$	$3$
$1$	$5$

2.

${\begin{cases} y = - 5 x - 9 \\ y = - 4 x + 2 \end{cases}$

3.

4.

What is the definition of a function?

For each graph determine if the relationship represents a function. If it is a function, write yes. If it is not a function, explain why it is not.