Precalculus

Robert Blitzer

Chapter 8

Matrices and Determinants - all with Video Answers

Educators

WZ

YJ

Section 1

Matrix Solutions to Linear Systems

Problem 1

In Exercises 1–8, write the augmented matrix for each system of linear equations.
$$\left\{\begin{aligned} 2 x+y+2 z=& 2 \\ 3 x-5 y-z=& 4 \\ x-2 y-3 z=&-6 \end{aligned}\right.$$

Joanna Quigley

Joanna Quigley

Numerade Educator

Problem 2

In Exercises 1–8, write the augmented matrix for each system of linear equations.
$$\left\{\begin{aligned} 3 x-2 y+5 z=& 31 \\ x+3 y-3 z=&-12 \\-2 x-5 y+3 z=& 11 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 3

In Exercises 1–8, write the augmented matrix for each system of linear equations.
$$\left\{\begin{aligned} x-y+z &=8 \\ y-12 z &=-15 \\ z &=1 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 4

In Exercises 1–8, write the augmented matrix for each system of linear equations.
$$\left\{\begin{aligned} x-2 y+3 z &=9 \\ y+3 z &=5 \\ z &=2 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 5

In Exercises 1–8, write the augmented matrix for each system of linear equations.
$$\left\{\begin{aligned} 5 x-2 y-3 z &=0 \\ x+y &=5 \\ 2 x-3 z &=4 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 6

In Exercises 1–8, write the augmented matrix for each system of linear equations.
$$\left\{\begin{aligned} x-2 y+z &=10 \\ 3 x+y &=5 \\ 7 x+2 z &=2 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 7

In Exercises 1–8, write the augmented matrix for each system of linear equations.
$$\left\{\begin{aligned} 2 w+5 x-3 y+z &=2 \\ 3 x+y &=4 \\ w-x+5 y &=9 \\ 5 w-5 x-2 y &=1 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 8

In Exercises 1–8, write the augmented matrix for each system of linear equations.
$$\left\{\begin{aligned} 4 w+7 x-8 y+z &=3 \\ 5 x+y &=5 \\ w-x-y &=17 \\ 2 w-2 x+11 y &=4 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 9

In Exercises $9-12,$ write the system of linear equations represented by the augmented matrix. Use $x, y,$ and $z,$ or, if necessary, $w, x, y,$ and z, for the variables.
$$\left[\begin{array}{rrr|r}{5} & {0} & {3} & {-11} \\ {0} & {1} & {-4} & {12} \\ {7} & {2} & {0} & {3}\end{array}\right]$$

Joanna Quigley

Numerade Educator

Problem 10

In Exercises $9-12,$ write the system of linear equations represented by the augmented matrix. Use $x, y,$ and $z,$ or, if necessary, $w, x, y,$ and z, for the variables.
$$\left[\begin{array}{rrr|r}{7} & {0} & {4} & {-13} \\ {0} & {1} & {-5} & {11} \\ {2} & {7} & {0} & {6}\end{array}\right]$$

Joanna Quigley

Numerade Educator

Problem 11

In Exercises $9-12,$ write the system of linear equations represented by the augmented matrix. Use $x, y,$ and $z,$ or, if necessary, $w, x, y,$ and z, for the variables.
$$\left[\begin{array}{rrrr|r}{1} & {1} & {4} & {1} & {3} \\ {-1} & {1} & {-1} & {0} & {7} \\ {2} & {0} & {0} & {5} & {11} \\ {0} & {0} & {12} & {4} & {5}\end{array}\right]$$

Joanna Quigley

Numerade Educator

Problem 12

In Exercises $9-12,$ write the system of linear equations represented by the augmented matrix. Use $x, y,$ and $z,$ or, if necessary, $w, x, y,$ and z, for the variables.
$$\left[\begin{array}{rrrr|r}{4} & {1} & {5} & {1} & {6} \\ {1} & {-1} & {0} & {-1} & {8} \\ {3} & {0} & {0} & {7} & {4} \\ {0} & {0} & {11} & {5} & {3}\end{array}\right]$$

Joanna Quigley

Numerade Educator

Problem 13

In Exercises $13-18,$ perform each matrix row operation and write the new matrix.
$$\left[\begin{array}{rrr|r}{2} & {-6} & {4} & {10} \\ {1} & {5} & {-5} & {0} \\ {3} & {0} & {4} & {7}\end{array}\right]^{\frac{1}{2} R_{1}}$$

Joanna Quigley

Numerade Educator

Problem 14

In Exercises $13-18,$ perform each matrix row operation and write the new matrix.
$$\left[\begin{array}{rrr|r}{3} & {-12} & {6} & {9} \\ {1} & {-4} & {4} & {0} \\ {2} & {0} & {7} & {4}\end{array}\right]$$

Joanna Quigley

Numerade Educator

Problem 15

In Exercises $13-18,$ perform each matrix row operation and write the new matrix.
$$\left[\begin{array}{rrr|r}{1} & {-3} & {2} & {0} \\ {3} & {1} & {-1} & {7} \\ {2} & {-2} & {1} & {3}\end{array}\right]-3 R_{1}+R_{2}$$

Joanna Quigley

Numerade Educator

Problem 16

In Exercises $13-18,$ perform each matrix row operation and write the new matrix.
$$\left[\begin{array}{rrr|r}{1} & {-1} & {5} & {-6} \\ {3} & {3} & {-1} & {10} \\ {1} & {3} & {2} & {5}\end{array}\right] \quad-3 R_{1}+R_{2}$$

Joanna Quigley

Numerade Educator

Problem 17

In Exercises $13-18,$ perform each matrix row operation and write the new matrix.
$$\left[\begin{array}{rrrr|r}{1} & {-1} & {1} & {1} & {3} \\ {0} & {1} & {-2} & {-1} & {0} \\ {2} & {0} & {3} & {4} & {11} \\ {5} & {1} & {2} & {4} & {6}\end{array}\right] \begin{array}{r}{-2 R_{1}+R_{3}} \\ {-5 R_{1}+R_{4}}\end{array}$$

Joanna Quigley

Numerade Educator

Problem 18

In Exercises $13-18,$ perform each matrix row operation and write the new matrix.
$$\left[\begin{array}{rrrr|r}{1} & {-5} & {2} & {-2} & {4} \\ {0} & {1} & {-3} & {-1} & {0} \\ {3} & {0} & {2} & {-1} & {6} \\ {-4} & {1} & {4} & {2} & {-3}\end{array}\right] \quad-3 R_{1}+R_{4}$$

Joanna Quigley

Numerade Educator

Problem 19

In Exercises $19-20,$ a few steps in the process of simplifying the given matrix to row-echelon form, with 1 s down the diagonal from upper left to lower right, and $0s$ below the $1s,$ are shown. Fill in the missing numbers in the steps that are shown.
$$\left[\begin{array}{rrr|r}{1} & {-1} & {1} & {8} \\ {2} & {3} & {-1} & {-2} \\ {3} & {-2} & {-9} & {9}\end{array}\right] \rightarrow\left[\begin{array}{rrr|r}{1} & {-1} & {1} & {8} \\ {0} & {5} & {\square} & {\square} \\ {0} & {1} & {\square} & {\square}\end{array}\right]$$
$$\qquad\qquad\qquad\qquad\qquad\rightarrow\left[\begin{array}{rrr|r}{1} & {-1} & {1} & {8} \\ {0} & {1} & {\square} & {\square} \\ {0} & {1} & {\square} & {\square}\end{array}\right]$$

Joanna Quigley

Numerade Educator

Problem 20

In Exercises $19-20,$ a few steps in the process of simplifying the given matrix to row-echelon form, with 1 s down the diagonal from upper left to lower right, and $0s$ below the $1s,$ are shown. Fill in the missing numbers in the steps that are shown.
$$\left[\begin{array}{rrr|r}{1} & {-2} & {3} & {4} \\ {2} & {1} & {-4} & {3} \\ {-3} & {4} & {-1} & {-2}\end{array}\right] \rightarrow\left[\begin{array}{rrr|r}{1} & {-2} & {3} & {4} \\ {0} & {5} & {\square} & {\square} \\ {0} & {-2} & {\square} & {\square}\end{array}\right]$$
$$\qquad\qquad\qquad\quad\qquad\qquad\rightarrow\left[\begin{array}{rrr|r}{1} & {-2} & {3} & {4} \\ {0} & {1} & {\square} & {\square} \\ {0} & {-2} & {\square} & {\square}\end{array}\right]$$

Joanna Quigley

Numerade Educator

Problem 21

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} x+y-z=&-2 \\ 2 x-y+z=& 5 \\-x+2 y+2 z=& 1 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 22

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} x-2 y-z=& 2 \\ 2 x-y+z=& 4 \\-x+y-2 z=&-4 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 23

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} x+3 y &=0 \\ x+y+z &=1 \\ 3 x-y-z &=11 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 24

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} 3 y-z &=-1 \\ x+5 y-z &=-4 \\-3 x+6 y+2 z &=11 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 25

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} 2 x-y-z=& 4 \\ x+y-5 z=&-4 \\ x-2 y &=4 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 26

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} x \quad\quad-3z=-2 \\ 2 x+2 y+z =4 \\ 3 x+y-2 z =5 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 27

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{array}{l}{x+y+z=4} \\ {x-y-z=0} \\ {x-y+z=2}\end{array}\right.$$

Joanna Quigley

Numerade Educator

Problem 28

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} 3 x+y-z=& 0 \\ x+y+2 z=& 6 \\ 2 x+2 y+3 z=& 10 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 29

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{array}{l}{x+2 y=z-1} \\ {x=4+y-z} \\ {x+y-3 z=-2}\end{array}\right.$$

Joanna Quigley

Numerade Educator

Problem 30

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{array}{c}{2 x+y=z+1} \\ {2 x=1+3 y-z} \\ {x+y+z=4}\end{array}\right.$$

Joanna Quigley

Numerade Educator

Problem 31

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{array}{r}{3 a-b-4 c=3} \\ {2 a-b+2 c=-8} \\ {a+2 b-3 c=9}\end{array}\right.$$

Joanna Quigley

Numerade Educator

Problem 32

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} 3 a+b-c=& 0 \\ 2 a+3 b-5 c=& 1 \\ a-2 b+3 c=&-4 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 33

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{array}{l}{2 x+2 y+7 z=-1} \\ {2 x+y+2 z=2} \\ {4 x+6 y+z=15}\end{array}\right.$$

Joanna Quigley

Numerade Educator

Problem 34

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{array}{l}{3 x+2 y+3 z=3} \\ {4 x-5 y+7 z=1} \\ {2 x+3 y-2 z=6}\end{array}\right.$$

Joanna Quigley

Numerade Educator

Problem 35

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} w+x+y+z=& 4 \\ 2 w+x-2 y-z=& 0 \\ w-2 x-y-2 z=&-2 \\ 3 w+2 x+y+3 z=& 4 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 36

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} w+x+y+z &=5 \\ w+2 x-y-2 z &=-1 \\ w-3 x-3 y-z &=-1 \\ 2 w-x+2 y-z &=-2 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 37

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} 3 w-4 x+y+z &=9 \\ w+x-y-z &=0 \\ 2 w+x+4 y-2 z &=3 \\-w+2 x+y-3 z &=3 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 38

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{aligned} 2 w+y-3 z &=8 \\ w-x+4 z &=-10 \\ 3 w+5 x-y-z &=20 \\ w+x-y-z &=6 \end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 39

Find the quadratic function $f(x)=a x^{2}+b x+c$ for which $f(-2)=-4, f(1)=2,$ and $f(2)=0.$

Joanna Quigley

Numerade Educator

Problem 40

Find the quadratic function $f(x)=a x^{2}+b x+c$ for which $f(-1)=5, f(1)=3,$ and $f(2)=5.$

Joanna Quigley

Numerade Educator

Problem 41

Find the cubic function $f(x)=a x^{3}+b x^{2}+c x+d$ for which $f(-1)=0, f(1)=2, f(2)=3,$ and $f(3)=12.$

Joanna Quigley

Numerade Educator

Problem 42

Find the cubic function $f(x)=a x^{3}+b x^{2}+c x+d$ for which $f(-1)=3, f(1)=1, f(2)=6,$ and $f(3)=7.$

Joanna Quigley

Numerade Educator

Problem 43

Solve the system:
$$\left\{\begin{aligned} 2 \ln w+\ln x+3 \ln y-2 \ln z &=-6 \\ 4 \ln w+3 \ln x+\ln y-\ln z &=-2 \\ \ln w+\ln x+\ln y+\ln z &=-5 \\ \ln w+\ln x-\ln y-\ln z &=5 \end{aligned}\right.$$
(Hint: Let $A=\ln w, B=\ln x, C=\ln y,$ and $D=\ln z .$ Solve the system for $A, B, C,$ and $D .$ Then use the logarithmic equations to find $w, x, y,$ and $z$ )

Joanna Quigley

Numerade Educator

Problem 44

Solve the system:
$$\left\{\begin{aligned} \ln w+\ln x+\ln y+\ln z &=-1 \\-\ln w+4 \ln x+\ln y-\ln z &=0 \\ \ln w-2 \ln x+\ln y-2 \ln z &=11 \\-\ln w-2 \ln x+\ln y+2 \ln z &=-3 \end{aligned}\right.$$
(Hint: Let $A=\ln w, B=\ln x, C=\ln y,$ and $D=\ln z$ . Solve the system for $A, B, C,$ and $D .$ Then use the logarithmic equations to find $w, x, y,$ and $z . )$

Joanna Quigley

Numerade Educator

Problem 45

A ball is thrown straight upward. A position function
$$s(t)=\frac{1}{2} a t^{2}+v_{0} t+s_{0}$$
can be used to describe the ball's height, $s(t),$ in feet, after $t$ seconds.
a. Use the points labeled in the graph to find the values of $a, v_{0},$ and $s_{0}$ . Solve the system of linear equations involving $a, v_{0},$ and $s_{0}$ using matrices.
b. Find and interpret $s(3.5)$ . Identify your solution as a point on the graph shown.
c. After how many scconds does the ball reach its maximum height? What is its maximum height?

Joanna Quigley

Numerade Educator

Problem 46

A football is kicked straight upward. A position function
$$s(t)=\frac{1}{2} a t^{2}+v_{0} t+s_{0}$$
can be used to describe the ball's height, $s(t),$ in feet, after $t$ seconds.
a. Use the points labeled in the graph to find the values of $a, v_{0},$ and $s_{0}$ . Solve the system of linear equations involving $a, v_{0}$ , and $s_{0}$ using matrices.
b. Find and interpret $s(7)$ . Identify your solution as a point on the graph shown.
c. After how many seconds does the ball reach its maximum height? What is its maximum height?

Joanna Quigley

Numerade Educator

Problem 47

Write a system of linear equations in three or four variables to solve Exercises 47–50. Then use matrices to solve the system.
Three foods have the following nutritional content per ounce.
$$\begin{array}{|l|l|l|l|} \hline \text {} & \text {Calories} & \text {Protein (in grams)} & \text {Vitamin C
(in milligrams)} \\ \hline {\text { Food } A} & {40} & {5} & {30} \\ \hline{\text { Food } B} & {200} & {2} & {10} \\ \hline{\text{ Food } C} & {400} & {4} & {300}\\ \hline \end{array} $$
If a meal consisting of the three foods allows exactly 660 calories, 25 grams of protein, and 425 milligrams of vitamin C, how many ounces of each kind of food should be used?

Joanna Quigley

Numerade Educator

Problem 48

A furniture company produces three types of desks: a children’s model, an office model, and a deluxe model. Each desk is manufactured in three stages: cutting, construction, and finishing. The time requirements for each model and manufacturing stage are given in the following table.
$$\begin{array}{|l|l|l|l|} \hline \text {} &\text {Children’s Model} &\text {Office Model} &\text {Deluxe Model}\\ \hline {\text { Cutting }} & {2 \mathrm{hr}} & {3 \mathrm{hr}} & {2 \mathrm{hr}} \\ \hline{\text { Construction }} & {2 \mathrm{hr}} & {1 \mathrm{hr}} & {3 \mathrm{hr}} \\ \hline{\text { Finishing }} & {1 \mathrm{hr}} & {1 \mathrm{hr}} & {2 \mathrm{hr}}\\ \hline\end{array}$$
Each week the company has available a maximum of 100 hours for cutting, 100 hours for construction, and 65 hours for finishing. If all available time must be used, how many of each type of desk should be produced each week?

Joanna Quigley

Numerade Educator

Problem 49

Imagine the entire global population as a village of precisely 200 people. The bar graph shows some numeric observations based on this scenario.
Combined, there are 183 Asians, Africans, Europeans, and Americans in the village. The number of Asians exceeds the number of Africans and Europeans by 70. The difference between the number of Europeans and Americans is 15. If the number of Africans is doubled, their population exceeds the number of Europeans and Americans by 23. Determine the number of Asians, Africans, Europeans, and Americans in the global village.

Joanna Quigley

Numerade Educator

Problem 50

The bar graph shows the number of rooms, bathrooms, fireplaces, and elevators in the U.S. White House.
Combined, there are 198 rooms, bathrooms, fireplaces, and elevators. The number of rooms exceeds the number of bathrooms and fireplaces by 69. The difference between the number of fireplaces and elevators is 25. If the number of bathrooms is doubled, it exceeds the number of fireplaces and elevators by 39. Determine the number of rooms, bathrooms, fireplaces, and elevators in the U.S. White House.

Joanna Quigley

Numerade Educator

Problem 51

What is a matrix?

Joanna Quigley

Numerade Educator

Problem 52

Describe what is meant by the augmented matrix of a system of linear equations.

Joanna Quigley

Numerade Educator

Problem 53

In your own words, describe each of the three matrix row operations. Give an example with each of the operations.

Joanna Quigley

Numerade Educator

Problem 54

Describe how to use row operations and matrices to solve a system of linear equations.

Joanna Quigley

Numerade Educator

Problem 55

What is the difference between Gaussian elimination and Gauss-Jordan elimination?

Joanna Quigley

Numerade Educator

Problem 56

Most graphing utilities can perform row operations on matrices. Consult the owner’s manual for your graphing utility to learn proper keystrokes for performing these operations. Then duplicate the row operations of any three exercises that you solved from Exercises 13–18.

Joanna Quigley

Numerade Educator

Problem 57

If your graphing utility has a REF (row-echelon form) command or a RREF (reduced row-echelon form) command, use this feature to verify your work with any five systems that you solved from Exercises 21–38.

Joanna Quigley

Numerade Educator

Problem 58

Solve using a graphing utility’s REF or RREF command:
$$\left\{\begin{aligned} 2 x_{1}-2 x_{2}+3 x_{3}-x_{4} =12 \\ x_{1}+2 x_{2}-x_{3}+2 x_{4}-x_{5} =-7 \\ x_{1}+ x_{3}+x_{4}-5 x_{5}=1 \\-x_{1}+x_{2}-x_{3}-2 x_{4}-3 x_{5} =0 \\ x_{1}-x_{2} x_{4}+x_{5}= 4\end{aligned}\right.$$

Joanna Quigley

Numerade Educator

Problem 59

In Exercises 59–62, determine whether each statement makes sense or does not make sense, and explain your reasoning.
Matrix row operations remind me of what I did when solving a linear system by the addition method, although I no longer write the variables.

Joanna Quigley

Numerade Educator

Problem 60

In Exercises 59–62, determine whether each statement makes sense or does not make sense, and explain your reasoning.
When I use matrices to solve linear systems, the only arithmetic involves multiplication or a combination of multiplication and addition.

Joanna Quigley

Numerade Educator

Problem 61

In Exercises 59–62, determine whether each statement makes sense or does not make sense, and explain your reasoning.
When I use matrices to solve linear systems, I spend most of my time using row operations to express the system’s augmented matrix in row-echelon form.

Joanna Quigley

Numerade Educator

Problem 62

In Exercises 59–62, determine whether each statement makes sense or does not make sense, and explain your reasoning.
Using row operations on an augmented matrix, I obtain a row in which 0s appear to the left of the vertical bar, but 6 appears on the right, so the system I’m working with has no solution.

Joanna Quigley

Numerade Educator

Problem 63

In Exercises 63–66, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
A matrix row operation such as $-\frac{4}{5} R_{1}+R_{2}$ is not permitted because of the negative fraction.

Joanna Quigley

Numerade Educator

Problem 64

The augmented matrix for the system
$\left\{\begin{array}{c}{x-3 y=5} \\ {y-2 z=7} \\ {2 x+z=4}\end{array}\ \text{is}\ \left[\begin{array}{cc|c}{1} & {-3} & {5} \\ {1} & {-2} & {7} \\ {2} & {1} & {4}\end{array}\right]\right.$

Joanna Quigley

Numerade Educator

Problem 65

In solving a linear system of three equations in three variables, we begin with the augmented matrix and use row operations to obtain a row-equivalent matrix with $0s$ down the diagonal from left to right and $1s$ below each 0.

Joanna Quigley

Numerade Educator

Problem 66

The row operation $k R_{i}+R_{j}$ indicates that it is the elements in row $i$ that change.

Joanna Quigley

Numerade Educator

Problem 67

The table shows the daily production level and profit for a business.
$$\begin{array}{|l|l|l|l|}\hline {\text { (Number of Units }} &{30} & {50} & {100} \\ {\text { Produced Daily }} \\ \hline {y \text { (Daily Profit) }} & {\$ 5900} & {\$ 7500} & {\$ 4500} \\ \hline \end{array}$$
Use the quadratic function $y=a x^{2}+b x+c$ to determine he number of units that should be produced each day for maximum profit. What is the maximum daily profit?

Joanna Quigley

Numerade Educator

Problem 68

Exercises 68–70 will help you prepare for the material covered in the next section. In each exercise, refer to the following system:
$$\left\{\begin{aligned} 3 x-4 y+4 z &=7 \\ x-y-2 z &=2 \\ 2 x-3 y+6 z &=5 \end{aligned}\right.$$
Show that $(12 z+1,10 z-1, z)$ satisfies the system for $z=0 .$

Joanna Quigley

Numerade Educator

Problem 69

Exercises 68–70 will help you prepare for the material covered in the next section. In each exercise, refer to the following system:
$$\left\{\begin{aligned} 3 x-4 y+4 z &=7 \\ x-y-2 z &=2 \\ 2 x-3 y+6 z &=5 \end{aligned}\right.$$
Show that $(12 z+1,10 z-1, z)$ satisfies the system for $z=1 .$

Joanna Quigley

Numerade Educator

Problem 70

Exercises 68–70 will help you prepare for the material covered in the next section. In each exercise, refer to the following system:
$$\left\{\begin{aligned} 3 x-4 y+4 z &=7 \\ x-y-2 z &=2 \\ 2 x-3 y+6 z &=5 \end{aligned}\right.$$
a. Select a value for $z$ other than 0 or 1 and show that $(12 z+1,10 z-1, z)$ satisfies the system.
b. Based on your work in Exercises $68-70(a),$ how does this system differ from those in Exercises $21-34 ?$

Joanna Quigley

Numerade Educator