Lesson 10 Taken Out of Context Practice Understanding

Jump Start

Your teacher will distribute a set of cards to you and your partner.

The “begin here” card gives you a system of equations.

Put the remaining cards in the order you would use to find the solution to the system.

On each card you will need to write an additional piece of information: (1) the missing equation on the top of the card, or (2) the missing justification for what you did to transition from the previous card to the current one.

Learning Focus

Solve systems of equations.

Identify systems that have no solutions or an infinite number of solutions.

Do all systems of linear equations have a solution?

Can a system of linear equations have more than one solution?

What features of a context help me think about the nature of the solution?

Open Up the Math: Launch, Explore, Discuss

Write a shopping scenario similar to those in Shopping for Cats and Dogs to fit each of the following systems of equations. Then use the elimination of variables method you invented in the previous lesson to solve the system. Some of the systems may have interesting or unusual solutions. See if you can explain them in terms of the shopping scenarios you wrote.

If you can’t find the solution to the system by elimination, try solving the system graphically or by substitution.

1.

${\begin{matrix} 3 x + 4 y = 23 \\ 5 x + 3 y = 31 \end{matrix}$

2.

${\begin{matrix} 2 x + 3 y = 14 \\ 4 x + 6 y = 28 \end{matrix}$

3.

${\begin{matrix} 3 x + 2 y = 20 \\ 9 x + 6 y = 36 \end{matrix}$

4.

${\begin{matrix} 4 x + 2 y = 8 \\ 5 x + 3 y = 9 \end{matrix}$

Pause and Reflect

5.

Three of Carlos and Clarita’s friends are purchasing school supplies at the bookstore. Stan buys a notebook, $3$ packages of pencils, and $2$ markers for $$ 7.50$ . Jan buys $2$ notebooks, $6$ packages of pencils, and $5$ markers for $$ 15.50$ . Fran buys $1$ notebook, $2$ packages of pencils, and $2$ markers for $$ 6.25$ . How much do each of these $3$ items cost?

The story context can be represented by this system of equations: ${\begin{matrix} n + 3 p + 2 m = 7.50 \\ 2 n + 6 p + 5 m = 15.50 \\ n + 2 p + 2 m = 6.25 \end{matrix}$

Explain in words or with symbols how you can use your intuitive reasoning about these purchases to find the price of each item.

Ready for More?

Create a linear system with three variables, $x$ , $y$ , and $z$ , by first deciding what you want the values of each variable to be and then writing three equations with random choices for the coefficients. Calculate the correct sums of the three equations using the known values of the variables.

Trade your system with a partner and see if you can solve the system that was created for you, and if your partner can solve the system that was created for them.

What features of a system make this easy? What makes it hard?

Takeaways

Describe the following systems of equations:

A dependent system:

An inconsistent system:

Finding a unique solution for the variables in an independent system of equations requires:

Vocabulary

systems: inconsistent / independent
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned that systems of linear equations can have infinitely many solutions, no solutions, or one solution, and we learned how to identify which type of solution to expect by examining the coefficients of the linear equations in the system when they are written in standard form. We also extended the process of solving systems to a system of linear equations with three unknowns.

Retrieval

Each of the pairs of equations below are equivalent. Determine what operation was performed on each term of the first equation to create the second equation.

1.

$9 x + 3 y = 27$

$3 x + y = 9$

2.

$- 5 x + 3 y = - 7$

$30 x - 18 y = 42$

3.

$4 x + 13 y = 50$

$13 y = - 4 x + 50$

Graph the following inequalities. Justify the region you shade by showing at least one point in the region as being a solution to each inequality.

4.

$- 6 x + 4 y \geq 12$

5.

$2 x + 6 y < 6$