Lesson 11E To Market with Matrices Solidify Understanding

Ready

Solve each system of equations using an algebraic method.

1.

${\begin{cases} 3 x - y = 1 \\ 3 x + 2 y = 16 \end{cases}$

2.

${\begin{cases} x + 2 y = 5 \\ 3 x + 5 y = 14 \end{cases}$

3.

${\begin{cases} 4 x + 2 y = - 8 \\ x - 2 y = - 7 \end{cases}$

4.

${\begin{cases} 3 x + 3 y = 2 \\ 3 x - 4 y = - 14 \end{cases}$

5.

${\begin{cases} x + 2 y = 11 \\ x - 4 y = 2 \end{cases}$

6.

${\begin{cases} 2 x + y = 0 \\ 5 x + 3 y = 1 \end{cases}$

Set

7.

Create a matrix to match each step in the solving of the system of equations given. Also, write a description of what happened to the equation and the matrix between steps.

System of Equations	Description	Matrix
Given System ${\begin{cases} 3 x + 2 y = 40 \\ x - 7 y = - 2 \end{cases}$ ⬇︎	Multiply row $2$ by $- 3$ ⬇︎	$[\begin{matrix} \begin{array}{r} 3 & 2 \\ 1 & - 7 \end{array} & \| \begin{array}{r} 40 \\ - 2 \end{array} \end{matrix}]$ ⬇︎
Step 1 ${\begin{cases} 3 x + 2 y = 40 \\ - 3 x + 21 y = 6 \end{cases}$ ⬇︎	⬇︎	$[\begin{matrix} \begin{matrix} \underset{―}{} & 2 \\ - 3 & \underset{―}{} \end{matrix} & \| \begin{array}{r} 40 \\ 6 \end{array} \end{matrix}]$ ⬇︎
Step 2 ${\begin{cases} 3 x + 2 y = 40 \\ 0 x + 23 y = 46 \end{cases}$ ⬇︎	⬇︎	$[\begin{matrix} \begin{matrix} \underset{―}{} & \underset{―}{} \\ 0 & \underset{―}{} \end{matrix} & \| \begin{array}{r} 40 \\ \underset{―}{} \end{array} \end{matrix}]$ ⬇︎
Step 3 ${\begin{cases} 3 x + 2 y = 40 \\ 0 x + y = 2 \end{cases}$ ⬇︎	⬇︎	$[\begin{matrix} \begin{array}{r} \underset{―}{} & \underset{―}{} \\ \underset{―}{} & \underset{―}{} \end{array} & \| \begin{array}{r} \underset{―}{} \\ \underset{―}{} \end{array} \end{matrix}]$ ⬇︎
Step 4 ${\begin{cases} 3 x + 0 y = 36 \\ 0 x + y = 2 \end{cases}$ ⬇︎		$[\begin{matrix} \begin{array}{r} \underset{―}{} & \underset{―}{} \\ \underset{―}{} & \underset{―}{} \end{array} & \| \begin{array}{r} \underset{―}{} \\ \underset{―}{} \end{array} \end{matrix}]$ ⬇︎
Step 5 ${\begin{cases} x + 0 y = 12 \\ 0 x + y = 2 \end{cases}$		$[\begin{matrix} \begin{matrix} \underset{―}{} & \underset{―}{} \\ \underset{―}{} & \underset{―}{} \end{matrix} & \| \begin{matrix} \underset{―}{} \\ \underset{―}{} \end{matrix} \end{matrix}]$

System of Equations

Description

Matrix

Given System ${\begin{cases} 3 x + 2 y = 40 \\ x - 7 y = - 2 \end{cases}$

⬇︎

Multiply row $2$ by $- 3$

⬇︎

$[\begin{matrix} \begin{array}{r} 3 & 2 \\ 1 & - 7 \end{array} & | \begin{array}{r} 40 \\ - 2 \end{array} \end{matrix}]$

⬇︎

Step 1 ${\begin{cases} 3 x + 2 y = 40 \\ - 3 x + 21 y = 6 \end{cases}$

⬇︎

$[\begin{matrix} \begin{matrix} \underset{―}{} & 2 \\ - 3 & \underset{―}{} \end{matrix} & | \begin{array}{r} 40 \\ 6 \end{array} \end{matrix}]$

⬇︎

Step 2 ${\begin{cases} 3 x + 2 y = 40 \\ 0 x + 23 y = 46 \end{cases}$

⬇︎

$[\begin{matrix} \begin{matrix} \underset{―}{} & \underset{―}{} \\ 0 & \underset{―}{} \end{matrix} & | \begin{array}{r} 40 \\ \underset{―}{} \end{array} \end{matrix}]$

⬇︎

Step 3 ${\begin{cases} 3 x + 2 y = 40 \\ 0 x + y = 2 \end{cases}$

⬇︎

$[\begin{matrix} \begin{array}{r} \underset{―}{} & \underset{―}{} \\ \underset{―}{} & \underset{―}{} \end{array} & | \begin{array}{r} \underset{―}{} \\ \underset{―}{} \end{array} \end{matrix}]$

⬇︎

Step 4 ${\begin{cases} 3 x + 0 y = 36 \\ 0 x + y = 2 \end{cases}$

⬇︎

Step 5 ${\begin{cases} x + 0 y = 12 \\ 0 x + y = 2 \end{cases}$

$[\begin{matrix} \begin{matrix} \underset{―}{} & \underset{―}{} \\ \underset{―}{} & \underset{―}{} \end{matrix} & | \begin{matrix} \underset{―}{} \\ \underset{―}{} \end{matrix} \end{matrix}]$

Go

Solve each system of equations by graphing.

8.

${\begin{cases} y = 3 x - 3 \\ y = - 3 x + 3 \end{cases}$

9.

${\begin{cases} y = 4 x - 1 \\ y = - x + 4 \end{cases}$

10.

${\begin{cases} y = - 2 x + 7 \\ - 3 x + y = - 8 \end{cases}$

11.

${\begin{cases} 4 x - y = 7 \\ 3 x + 2 y = 8 \end{cases}$