Lesson 6 All Systems Go Practice Understanding

Jump Start

You worked on the following problem as the Exit Ticket for the previous lesson. Today, you will have an opportunity to see all of the methods for solving this problem that were used by your peers.

Solve the following system of two equations with two variables using any one of the following strategies: $\begin{matrix} - 4 x + 3 y = 8 \\ 4 x - 2 y = 4 \end{matrix}$

Graphically
Using substitution
Using elimination of variables
Using matrices and row reduction (for example, getting $1$ ’s and $0$ ’s in appropriate places in the matrix by combining rows or multiplying rows by a constant)
Using a matrix equation and the multiplicative inverse

Learning Focus

Solve $n \times n$ systems of linear equations.

What is the most efficient way to solve systems of equations that involve $n$ variables and $n$ linear equations?

Open Up the Math: Launch, Explore, Discuss

Carlos and Clarita are trying to figure out the cost of a gallon of paint, the cost of a paintbrush, and the cost of a roll of masking tape based on the following purchases:

Week 1: Carlos bought $2$ gallons of paint and $1$ roll of masking tape for $$ 30$ .

Week 2: Carlos bought $1$ gallon of paint and $4$ brushes for $$ 20$ .

Week 3: Carlos bought $2$ brushes and $1$ roll of masking tape for $$ 10$ .

In the previous activity, Solving Systems with Matrices, Revisited, you examined a method for solving systems using a matrix equation and the multiplicative inverse. In this task, we are going to extend this strategy to include systems with more than two equations and two variables.

1.

Multiply the following pairs of matrices:

a.

$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \cdot [\begin{matrix} 2 & 0 & 1 \\ 1 & 4 & 0 \\ 0 & 2 & 1 \end{matrix}]$

b.

$[\begin{matrix} 0.4 & 0.2 & - 0.4 \\ - 0.1 & 0.2 & 0.1 \\ 0.2 & - 0.4 & 0.8 \end{matrix}] \cdot [\begin{matrix} 2 & 0 & 1 \\ 1 & 4 & 0 \\ 0 & 2 & 1 \end{matrix}]$

2.

What property is illustrated by the multiplication in problem 1a?

3.

What property is illustrated by the multiplication in problem 1b?

4.

Rewrite the following system of equations, which represents Carlos’s and Clarita’s problem, as a matrix equation in the form $A X = B$ where $A$ , $X$ , and $B$ are all matrices.

$\begin{matrix} 2 g + 0 b + 1 t = 30 \\ 1 g + 4 b + 0 t = 20 \\ 0 g + 2 b + 1 t = 10 \end{matrix}$

5.

Solve your matrix equation by using multiplication of matrices. Show the details of your work so that someone else can follow it.

Pause and Reflect

You were able to solve this equation using matrix multiplication because you were given the inverse of matrix $A$ . Unlike $2 \times 2$ matrices, where the inverse matrix can easily be found by hand using the methods described in earlier lessons, the inverses of an $n \times n$ matrix in general can be difficult to find by hand. In such cases, we will use technology to find the inverse matrix so that this method can be applied to all linear systems involving $n$ equations and $n$ unknown quantities. Here is one online resource you might use: https://openup.org/XsBpit.

6.

$\begin{matrix} x + 2 y + 2 z = 16 \\ 2 x + y + 3 z = 17 \\ x + 2 y + z = 13 \end{matrix}$

7.

Solve the following system using a matrix equation and inverse matrices. Although you may use technology to find the inverse matrix, make sure you record all of your work in the space below, including your inverse matrix. (Note: If you worked on the Ready for More in the previous task, you have already found the multiplicative inverse of the matrix you will use to solve this system.)

$\begin{matrix} 2 x + z = 7 \\ x + 3 y + z = 11 \\ x + 2 y + z = 6 \end{matrix}$

8.

Solve the following system using a matrix equation and inverse matrices. Use technology to find the inverse matrix, and the product of the matrices that will produce the answer:

$\begin{matrix} 2 w + x + y + 2 z = 7 \\ w + 2 x + y + 3 z = 10 \\ \begin{matrix} 2 w - x + 2 y + 5 z = - 2 \\ 3 w + 2 x + 2 y + z = 5 \end{matrix} \end{matrix}$

9.

Why do you need four equations when there are four variables?

10.

Solve the following system using technology:

$\begin{matrix} 2 w + x + y + 2 z = 7 \\ 4 w + 2 x + 2 y + 4 z = 14 \\ \begin{matrix} 2 w - x + 2 y + 5 z = - 2 \\ 4 w - 2 x + 4 y + 10 z = - 4 \end{matrix} \end{matrix}$

Ready for More?

Use technology to solve the following problem:

Three of Carlos’s and Clarita’s friends are purchasing school supplies at the bookstore. Stan buys $1 notebook$ , $3$ 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?

Takeaways

To solve a system of equations in $n$ variables you need

If the determinant of a matrix is $0$ ,

The preferred method for solving $n \times n$ linear systems is

This method is preferred because

Lesson Summary

In this lesson, we extended the method of solving systems by using matrix equations and the multiplicative inverse to $n \times n$ systems of equations. We used technology to find the multiplicative inverse matrix.

Retrieval

1.

Write the exponential expression in radical form.

${(5 x^{4})}^{\frac{1}{3}}$

2.

Write the radical expression in exponential form.

$\sqrt[7]{13 a^{5} b^{3}}$

3.

Solve the quadratic equation using an efficient method.

${(x - 6)}^{2} - 8 = 0$

4.

Solve the quadratic equation using an efficient method.

$3 x^{2} - 12 x + 1 = 0$