Algebra and Trigonometry

Cynthia Y. Young

Chapter 10

Matrices - all with Video Answers

Educators

Section 1

MATRICES AND SYSTEMS OF LINEAR EOUATIONS

Problem 1

In Exercises 1-6, determine the order of each matrix.
$$
\left[\begin{array}{rrr}
-1 & 3 & 4 \\
2 & 7 & 9
\end{array}\right]
$$

James Kiss

James Kiss

Numerade Educator

Problem 2

Determine the order of each matrix.
$$
\left[\begin{array}{ll}
0 & 1 \\
3 & 9 \\
7 & 8
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 3

Determine the order of each matrix.
$$
\left[\begin{array}{llll}
1 & 2 & 3 & 4
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 4

Determine the order of each matrix.
$$
\left[\begin{array}{r}
3 \\
7 \\
-1 \\
10
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 5

Determine the order of each matrix.
$$
[0]
$$

Thomas Emment

Numerade Educator

Problem 6

Determine the order of each matrix.
$$
\left[\begin{array}{rrrr}
-1 & 3 & 6 & 8 \\
2 & 9 & 7 & 3 \\
5 & 4 & -2 & -10 \\
6 & 3 & 1 & 5
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 7

In Exercises $7-14,$ write the augmented matrix for each system of linear equations.
$$
\begin{array}{r}
3 x-2 y=7 \\
-4 x+6 y=-3
\end{array}
$$

James Kiss

Numerade Educator

Problem 8

Write the augmented matrix for each system of linear equations.
$$
\begin{array}{r}
-x+y=2 \\
x-y=-4
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 9

Write the augmented matrix for each system of linear equations.
$$
\begin{aligned}
2 x-3 y+4 z &=-3 \\
-x+y+2 z &=1 \\
5 x-2 y-3 z &=7
\end{aligned}
$$

James Kiss

Numerade Educator

Problem 10

Write the augmented matrix for each system of linear equations.
$$
\begin{array}{rr}
x-2 y+z= & 0 \\
-2 x+y-z= & -5 \\
13 x+7 y+5 z= & 6
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 11

Write the augmented matrix for each system of linear equations.
$$
\begin{array}{l}
x+y=3 \\
x-z=2 \\
y+z=5
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 12

Write the augmented matrix for each system of linear equations.
$$
\begin{array}{l}
x-y=-4 \\
y+z=3
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 13

Write the augmented matrix for each system of linear equations.
$$
\begin{aligned}
3 y-4 x+5 z-2 &=0 \\
2 x-3 y-2 z &=-3 \\
3 z+4 y-2 x-1 &=0
\end{aligned}
$$

Thomas Emment

Numerade Educator

Problem 14

Write the augmented matrix for each system of linear equations.
$$
\begin{array}{r}
2 y+z-x-3=2 \\
2 x+3 z-2 y=0 \\
-2 z+y-4 x-3=0
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 15

In Exercises $15-20,$ write the system of linear equations represented by the augmented matrix. Utilize the variables $x, y$, and $z$.
$$
\left[\begin{array}{rr|r}
-3 & 7 & 2 \\
1 & 5 & 8
\end{array}\right]
$$

James Kiss

Numerade Educator

Problem 16

Write the system of linear equations represented by the augmented matrix. Utilize the variables $x, y$, and $z$.
$$
\left[\begin{array}{rrr|r}
-1 & 2 & 4 & 4 \\
7 & 9 & 3 & -3 \\
4 & 6 & -5 & 8
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 17

Write the system of linear equations represented by the augmented matrix. Utilize the variables $x, y$, and $z$.
$$
\left[\begin{array}{rrr|r}
-1 & 0 & 0 & 4 \\
7 & 9 & 3 & -3 \\
4 & 6 & -5 & 8
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 18

Write the system of linear equations represented by the augmented matrix. Utilize the variables $x, y$, and $z$.
$$
\left[\begin{array}{rrr|r}
2 & 3 & -4 & 6 \\
7 & -1 & 5 & 9
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 19

Write the system of linear equations represented by the augmented matrix. Utilize the variables $x, y$, and $z$.
$$
\left[\begin{array}{ll|l}
1 & 0 & a \\
0 & 1 & b
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 20

Write the system of linear equations represented by the augmented matrix. Utilize the variables $x, y$, and $z$.
$$
\left[\begin{array}{rrr|r}
3 & 0 & 5 & 1 \\
0 & -4 & 7 & -3 \\
2 & -1 & 0 & 8
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 21

In Exercises 21-30, indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$
\left[\begin{array}{ll|l}
1 & 0 & 3 \\
1 & 1 & 2
\end{array}\right]
$$

James Kiss

Numerade Educator

Problem 22

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$
\left[\begin{array}{ll|l}
0 & 1 & 3 \\
1 & 0 & 2
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 23

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$
\left[\begin{array}{rrr|r}
1 & 0 & -1 & -3 \\
0 & 1 & 3 & 14
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 24

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$
\left[\begin{array}{lll|r}
1 & 0 & 0 & -3 \\
0 & 1 & 3 & 14
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 25

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$
\left[\begin{array}{lll|l}
1 & 0 & 1 & 3 \\
0 & 0 & 0 & 0 \\
0 & 1 & 2 & 2
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 26

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$
\left[\begin{array}{lll|l}
1 & 0 & 1 & 3 \\
0 & 1 & 2 & 2 \\
0 & 0 & 0 & 0
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 27

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$
\left[\begin{array}{lll|l}
1 & 0 & 0 & 3 \\
0 & 1 & 0 & 2 \\
0 & 0 & 1 & 5
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 28

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$
\left[\begin{array}{rrr|r}
-1 & 0 & 0 & 3 \\
0 & -1 & 0 & 2 \\
0 & 0 & -1 & 5
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 29

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$
\left[\begin{array}{llll|l}
1 & 0 & 0 & 1 & 3 \\
0 & 1 & 0 & 3 & 2 \\
0 & 0 & 1 & 0 & 5 \\
0 & 0 & 0 & 1 & 0
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 30

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$
\left[\begin{array}{llll|l}
1 & 0 & 0 & 1 & 3 \\
0 & 1 & 0 & 3 & 2 \\
0 & 0 & 1 & 0 & 5 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 31

In Exercises $31-40,$ perform the indicated row operations on each augmented matrix.
$$
\left[\begin{array}{rr|r}
1 & -2 & -3 \\
2 & 3 & -1
\end{array}\right] \quad R_{2}-2 R_{1} \rightarrow R_{2}
$$

James Kiss

Numerade Educator

Problem 32

Perform the indicated row operations on each augmented matrix.
$$
\left[\begin{array}{rr|r}
2 & -3 & -4 \\
1 & 2 & 5
\end{array}\right] \quad R_{1} \leftrightarrow R_{2}
$$

Thomas Emment

Numerade Educator

Problem 33

Perform the indicated row operations on each augmented matrix.
$$
\left[\begin{array}{rrr|r}
1 & -2 & -1 & 3 \\
2 & 1 & -3 & 6 \\
3 & -2 & 5 & -8
\end{array}\right] R_{2}-2 R_{1} \rightarrow R_{2}
$$

Thomas Emment

Numerade Educator

Problem 34

Perform the indicated row operations on each augmented matrix.
$$
\left[\begin{array}{rrr|r}
1 & -2 & 1 & 3 \\
0 & 1 & -2 & 6 \\
-3 & 0 & -1 & -5
\end{array}\right] \quad R_{3}+3 R_{1} \rightarrow R_{3}
$$

Thomas Emment

Numerade Educator

Problem 35

Perform the indicated row operations on each augmented matrix.
$$
\left[\begin{array}{rrrr|r}
1 & -2 & 5 & -1 & 2 \\
0 & 3 & 0 & -1 & -2 \\
0 & -2 & 1 & -2 & 5 \\
0 & 0 & 1 & -1 & -6
\end{array}\right] R_{3}+R_{2} \rightarrow R_{2}
$$

Thomas Emment

Numerade Educator

Problem 36

Perform the indicated row operations on each augmented matrix.
$$
\left[\begin{array}{rrrr|r}
1 & 0 & 5 & -10 & 15 \\
0 & 1 & 2 & -3 & 4 \\
0 & 2 & -3 & 0 & -1 \\
0 & 0 & 1 & -1 & -3
\end{array}\right] R_{2}-\frac{1}{2} R_{3} \rightarrow R_{3}
$$

Thomas Emment

Numerade Educator

Problem 37

Perform the indicated row operations on each augmented matrix.
$$
\left[\begin{array}{rrrr|r}
1 & 0 & 5 & -10 & -5 \\
0 & 1 & 2 & -3 & -2 \\
0 & 2 & -3 & 0 & -1 \\
0 & -3 & 2 & -1 & -3
\end{array}\right] \quad \begin{array}{l}
R_{3}-2 R_{2} \rightarrow R_{3} \\
R_{4}+3 R_{2} \rightarrow R_{4}
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 38

Perform the indicated row operations on each augmented matrix.
$$
\left[\begin{array}{llll|r}
1 & 0 & 4 & 0 & 1 \\
0 & 1 & 2 & 0 & -2 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & -3
\end{array}\right] \quad \begin{array}{l}
R_{2}-2 R_{3} \rightarrow R_{2} \\
R_{1}-4 R_{3} \rightarrow R_{1}
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 39

Perform the indicated row operations on each augmented matrix.
$$
\left[\begin{array}{rrrr|r}
1 & 0 & 4 & 8 & 3 \\
0 & 1 & 2 & -3 & -2 \\
0 & 0 & 1 & 6 & 3 \\
0 & 0 & 0 & 1 & -3
\end{array}\right] \begin{array}{c}
R_{3}-6 R_{4} \rightarrow R_{3} \\
R_{2}+3 R_{4} \rightarrow R_{2} \\
R_{1}-8 R_{4} \rightarrow R_{1}
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 40

Perform the indicated row operations on each augmented matrix.
$$
\left[\begin{array}{rrrr|r}
1 & 0 & -1 & 5 & 2 \\
0 & 1 & 2 & 3 & -5 \\
0 & 0 & 1 & -2 & 2 \\
0 & 0 & 0 & 1 & 1
\end{array}\right] \begin{array}{c}
R_{3}+2 R_{4} \rightarrow R_{3} \\
R_{2}-3 R_{4} \rightarrow R_{2} \\
R_{1}-5 R_{4} \rightarrow R_{1}
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 41

In Exercises 41-50, use row operations to transform each matrix to reduced row-echelon form.
$$
\left[\begin{array}{ll|l}
1 & 2 & 4 \\
2 & 3 & 2
\end{array}\right]
$$

James Kiss

Numerade Educator

Problem 42

Use row operations to transform each matrix to reduced row-echelon form.
$$
\left[\begin{array}{rr|r}
1 & -1 & 3 \\
-3 & 2 & 2
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 43

Use row operations to transform each matrix to reduced row-echelon form.
$$
\left[\begin{array}{rrr|r}
1 & -1 & 1 & -1 \\
0 & 1 & -1 & -1 \\
-1 & 1 & 1 & 1
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 44

Use row operations to transform each matrix to reduced row-echelon form.
$$
\left[\begin{array}{rrr|r}
0 & -1 & 1 & 1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & -1
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 45

Use row operations to transform each matrix to reduced row-echelon form.
$$
\left[\begin{array}{rrr|r}
3 & -2 & -3 & -1 \\
1 & -1 & 1 & -4 \\
2 & 3 & 5 & 14
\end{array}\right]
$$

James Kiss

Numerade Educator

Problem 46

Use row operations to transform each matrix to reduced row-echelon form.
$$
\left[\begin{array}{rrr|r}
3 & -1 & 1 & 2 \\
1 & -2 & 3 & 1 \\
2 & 1 & -3 & -1
\end{array}\right]
$$

James Kiss

Numerade Educator

Problem 47

Use row operations to transform each matrix to reduced row-echelon form.
$$
\left[\begin{array}{rrr|r}
2 & 1 & -6 & 4 \\
1 & -2 & 2 & -3
\end{array}\right]
$$

Thomas Emment

Numerade Educator

Problem 48

Use row operations to transform each matrix to reduced row-echelon form.
$$
\left[\begin{array}{rrr|r}
-3 & -1 & 2 & -1 \\
-1 & -2 & 1 & -3
\end{array}\right]
$$

James Kiss

Numerade Educator

Problem 49

Use row operations to transform each matrix to reduced row-echelon form.
$$
\left[\begin{array}{rrr|r}
-1 & 2 & 1 & -2 \\
3 & -2 & 1 & 4 \\
2 & -4 & -2 & 4
\end{array}\right]
$$

James Kiss

Numerade Educator

Problem 50

Use row operations to transform each matrix to reduced row-echelon form.
$$
\left[\begin{array}{rrr|r}
2 & -1 & 0 & 1 \\
-1 & 0 & 1 & -2 \\
-2 & 1 & 0 & -1
\end{array}\right]
$$

James Kiss

Numerade Educator

Problem 51

In Exercises $51-70,$ solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{r}
2 x+3 y=1 \\
x+y=-2
\end{array}
$$

James Kiss

Numerade Educator

Problem 52

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{r}
3 x+2 y=11 \\
x-y=12
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 53

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{aligned}
-x+2 y &=3 \\
2 x-4 y &=-6
\end{aligned}
$$

Thomas Emment

Numerade Educator

Problem 54

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{l}
3 x-y=-1 \\
2 y+6 x=2
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 55

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{l}
\frac{2}{3} x+\frac{1}{3} y=\frac{8}{9} \\
\frac{1}{2} x+\frac{1}{4} y=\frac{3}{4}
\end{array}
$$

James Kiss

Numerade Educator

Problem 56

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{r}
0.4 x-0.5 y=2.08 \\
-0.3 x+0.7 y=1.88
\end{array}
$$

James Kiss

Numerade Educator

Problem 57

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{rr}
x-z-y= & 10 \\
2 x-3 y+z= & -11 \\
y-x+z= & -10
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 58

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{aligned}
2 x+z+y &=-3 \\
2 y-z+x &=0 \\
x+y+2 z &=5
\end{aligned}
$$

James Kiss

Numerade Educator

Problem 59

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{rr}
3 x_{1}+x_{2}-x_{3}= & 1 \\
x_{1}-x_{2}+x_{3}= & -3 \\
2 x_{1}+x_{2}+x_{3}= & 0
\end{array}
$$

James Kiss

Numerade Educator

Problem 60

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{r}
2 x_{1}+x_{2}+x_{3}=-1 \\
x_{1}+x_{2}-x_{3}=5 \\
3 x_{1}-x_{2}-x_{3}=1
\end{array}
$$

James Kiss

Numerade Educator

Problem 61

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{r}
2 x+5 y=9 \\
x+2 y-z=3 \\
-3 x-4 y+7 z=1
\end{array}
$$

James Kiss

Numerade Educator

Problem 62

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{r}
x-2 y+3 z=1 \\
-2 x+7 y-9 z=4 \\
x+z=9
\end{array}
$$

James Kiss

Numerade Educator

Problem 63

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{r}
2 x_{1}-x_{2}+x_{3}=3 \\
x_{1}-x_{2}+x_{3}=2 \\
-2 x_{1}+2 x_{2}-2 x_{3}=-4
\end{array}
$$

James Kiss

Numerade Educator

Problem 64

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{aligned}
x_{1}-x_{2}-2 x_{3} &=0 \\
-2 x_{1}+5 x_{2}+10 x_{3} &=-3 \\
3 x_{1}+x_{2} &=0
\end{aligned}
$$

James Kiss

Numerade Educator

Problem 65

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{r}
2 x+y-z=2 \\
x-y-z=6
\end{array}
$$

James Kiss

Numerade Educator

Problem 66

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{r}
3 x+y-z=0 \\
x+y+7 z=4
\end{array}
$$

Andrei Demkov

Numerade Educator

Problem 67

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{rr}
2 y+z= & 3 \\
4 x-z= & -3 \\
7 x-3 y-3 z= & 2 \\
x-y-z= & -2
\end{array}
$$

Andrei Demkov

Numerade Educator

Problem 68

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{aligned}
-2 x-y+2 z &=3 \\
3 x-4 z &=2 \\
2 x+y &=-1 \\
-x+y-z &=-8
\end{aligned}
$$

James Kiss

Numerade Educator

Problem 69

$$
\begin{array}{rr}
3 x_{1}-2 x_{2}+x_{3}+2 x_{4} & =-2 \\
-x_{1}+3 x_{2}+4 x_{3}+3 x_{4} & =4 \\
x_{1}+x_{2}+x_{3}+x_{4} & =0 \\
5 x_{1}+3 x_{2}+x_{3}+2 x_{4} & =-1
\end{array}
$$

Matthew Allcock

Matthew Allcock

Numerade Educator

Problem 69

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{rr}
3 x_{1}-2 x_{2}+x_{3}+2 x_{4}= & -2 \\
-x_{1}+3 x_{2}+4 x_{3}+3 x_{4}= & 4 \\
x_{1}+x_{2}+x_{3}+x_{4}= & 0 \\
5 x_{1}+3 x_{2}+x_{3}+2 x_{4}= & -1
\end{array}
$$

James Kiss

Numerade Educator

Problem 70

Solve the system of linear equations using Gaussian elimination with back-substitution.
$$
\begin{array}{rr}
5 x_{1}+3 x_{2}+8 x_{3}+x_{4}= & 1 \\
x_{1}+2 x_{2}+5 x_{3}+2 x_{4}= & 3 \\
4 x_{1}+x_{3}-2 x_{4}= & -3 \\
x_{2}+x_{3}+x_{4}= & 0
\end{array}
$$

James Kiss

Numerade Educator

Problem 71

In Exercises $71-86,$ solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{aligned}
x+3 y &=-5 \\
-2 x-y &=0
\end{aligned}
$$

James Kiss

Numerade Educator

Problem 72

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{array}{l}
5 x-4 y=31 \\
3 x+7 y=-19
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 73

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{array}{r}
x+y=4 \\
-3 x-3 y=10
\end{array}
$$

Thomas Emment

Numerade Educator

Problem 74

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{aligned}
3 x-4 y &=12 \\
-6 x+8 y &=-24
\end{aligned}
$$

Thomas Emment

Numerade Educator

Problem 75

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{array}{rr}
x-2 y+3 z= & 5 \\
3 x+6 y-4 z= & -12 \\
-x-4 y+6 z= & 16
\end{array}
$$

James Kiss

Numerade Educator

Problem 76

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{array}{r}
x+2 y-z=6 \\
2 x-y+3 z=-13 \\
3 x-2 y+3 z=-16
\end{array}
$$

James Kiss

Numerade Educator

Problem 77

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{array}{rr}
x+y+z= & 3 \\
x-z= & 1 \\
y-z= & -4
\end{array}
$$

James Kiss

Numerade Educator

Problem 78

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{aligned}
x-2 y+4 z &=2 \\
2 x-3 y-2 z &=-3 \\
\frac{1}{2} x+\frac{1}{4} y+z &=-2
\end{aligned}
$$

James Kiss

Numerade Educator

Problem 79

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{array}{r}
x+2 y+z=3 \\
2 x-y+3 z=7 \\
3 x+y+4 z=5
\end{array}
$$

James Kiss

Numerade Educator

Problem 80

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{aligned}
x+2 y+z &=3 \\
2 x-y+3 z &=7 \\
3 x+y+4 z &=10
\end{aligned}
$$

James Kiss

Numerade Educator

Problem 81

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{array}{r}
3 x-y+z=8 \\
x+y-2 z=4
\end{array}
$$

James Kiss

Numerade Educator

Problem 82

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{aligned}
x-2 y+3 z &=10 \\
-3 x+z &=9
\end{aligned}
$$

James Kiss

Numerade Educator

Problem 83

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{aligned}
4 x-2 y+5 z &=20 \\
x+3 y-2 z &=6
\end{aligned}
$$

James Kiss

Numerade Educator

Problem 84

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{array}{l}
y+z=4 \\
x+y=8
\end{array}
$$

James Kiss

Numerade Educator

Problem 85

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{array}{rr}
x-y-z-w= & 1 \\
2 x+y+z+2 w= & 3 \\
x-2 y-2 z-3 w= & 0 \\
3 x-4 y+z+5 w= & -3
\end{array}
$$

James Kiss

Numerade Educator

Problem 86

Solve the system of linear equations using Gauss-Jordan elimination.
$$
\begin{aligned}
x-3 y+3 z-2 w &=4 \\
x+2 y-z &=-3 \\
x+3 z+2 w &=3 \\
y+z+5 w &=6
\end{aligned}
$$

James Kiss

Numerade Educator

Problem 87

In Super Bowl XXXVIII, the New England Patriots defeated the Carolina Panthers $32-29 .$ The points came from a total of four types of plays: touchdowns ( 6 points), extra points (1 point), two-point conversions (2 points), and field goals $(3$ points $)$. There were a total of 16 scoring plays. There were four times as many touchdowns as field goals, and there were five times as many extra points as 2 -point conversions. How many touchdowns, extra points, 2 -point conversions, and field goals were scored in Super Bowl XXXVIII?

Marcella Sippey

Marcella Sippey

Numerade Educator

Problem 88

In the 2004 Summer Olympics in Athens, Greece, the U.S. men's basketball team, consisting of NBA superstars, was defeated by the Puerto Rican team $92-73$. The points came from three types of scoring plays: 2 -point shots, 3-point shots, and 1-point free throws. There were six more 2 -point shots made than there were 1 -point free throws. The number of successful 2 -point shots was three less than four times the number of successful 3 -point shots. How many 2 -point and 3 -point shots, and 1 -point free throws were made in that Olympic competition?

Jim Long

Numerade Educator

Problem 89

Exercises 89 and 90 rely on a selection of sandwiches whose nutrition information is given in the following table. Suppose you are going to eat only sandwiches for a week (7 days) for lunch and dinner (a total of 14 meals).
Your goal is a low-fat diet consisting of 526 grams of carbohydrates, 168 grams of fat, and 332 grams of protein. How many of each sandwich would you eat that week to obtain this goal?

Jim Long

Numerade Educator

Problem 90

Rely on a selection of sandwiches whose nutrition information is given in the following table. Suppose you are going to eat only sandwiches for a week (7 days) for lunch and dinner (a total of 14 meals).
Your goal is a low-carb diet consisting of 5180 calories, 335 grams of carbohydrates, and 263 grams of fat. How many of each sandwich would you eat that week to obtain this goal?

Abhishek Kumar

Numerade Educator

Problem 91

Exercises 91 and 92 involve vertical motion and the effect of gravity on an object.

Because of gravity, an object that is projected upward will eventually reach a maximum height and then fall to the ground. The equation that relates the height $h$ of a projectile $t$ seconds after it is shot upward is given by
$$
h=\frac{1}{2} a t^{2}+v_{0} t+h_{0}
$$
where $a$ is the acceleration due to gravity, $h_{0}$ is the initial height of the object at time $t=0,$ and $v_{0}$ is the initial velocity of the object at time $t=0 .$ Note that a projectile follows the path of a parabola opening down, so $a<0$.
An object is thrown upward, and the following table depicts the height of the ball $t$ seconds after the projectile is released. Find the initial height, initial velocity, and acceleration due to gravity.

Ashley Volpe

Numerade Educator

Problem 92

Involve vertical motion and the effect of gravity on an object. Because of gravity, an object that is projected upward will eventually reach a maximum height and then fall to the ground. The equation that relates the height $h$ of a projectile $t$ seconds after it is shot upward is given by
$$
h=\frac{1}{2} a t^{2}+v_{0} t+h_{0}
$$
where $a$ is the acceleration due to gravity, $h_{0}$ is the initial height of the object at time $t=0,$ and $v_{0}$ is the initial velocity of the object at time $t=0 .$ Note that a projectile follows the path of a parabola opening down, so $a<0$.
An object is thrown upward, and the following table depicts the height of the ball $t$ seconds after the projectile is released. Find the initial height, initial velocity and acceleration due to gravity.

Ashley Volpe

Numerade Educator

Problem 93

The average number of minutes that a person spends driving a car can be modeled by a quadratic function $y=a x^{2}+b x+c,$ where $a<0$ and $15<x<65$ The following table gives the average number of minutes a day that a person spends driving a car. Determine the quadratic function that models this quantity.

Ashley Volpe

Numerade Educator

Problem 94

The average age when a woman gets married has been increasing during the last century. In 1920 the average age was 18.4 , in 1960 the average age was 20.3 , and in 2002 the average age was 25.30 . Find a quadratic function $y=a x^{2}+b x+c,$ where $a>0$ and $18<x<35,$ that models the average age $y$ when a woman gets married as a function of the year $x(x=0$ corresponds to 1920$)$. What will the average age be in $2020 ?$

Ashley Volpe

Numerade Educator

Problem 95

A pharmacy receives an order for 100 milliliters of $5 \%$ hydrogen peroxide solution. The pharmacy has a $1.5 \%$ and a $30 \%$ solution on hand. A technician will mix the $1.5 \%$ and $30 \%$ solutions to make the $5 \%$ solution. How much of the $1.5 \%$ and $30 \%$ solutions, respectively, will be needed to fill this order? Round to the nearest ml.

Suzanne W.

Numerade Educator

Problem 96

A pharmacy receives an order for 60 grams of a $0.7 \%$ hydrocortisone cream. The pharmacy has $1 \%$ and $0.5 \%$ hydrocortisone creams as well as a Eucerin cream for use as a base (0\% hydrocortisone). The technician must use twice as much $0.5 \%$ hydrocortisone cream than the Eucerin base. How much of the $1 \%$ and $0.5 \%$ hydrocortisone creams and Eucerin cream are needed to fill this order?

Chelsea Green

Numerade Educator

Problem 97

A small company has an assembly line that produces three types of widgets. The basic widget is sold for $\$ 12$ per unit, the midprice widget for $\$ 15$ per unit, and the top-of-the-line widget for $\$ 18$ per unit. The assembly line has a daily capacity of producing 375 widgets that may be sold for a total of $\$ 5250$. Find the quantity of each type of widget produced on a day when twice as many basic widgets as midprice widgets are produced.

Jim Long

Numerade Educator

Problem 98

A small company has an assembly line that produces three types of widgets. The basic widget is sold for $\$ 10$ per unit, the midprice widget for $\$ 12$ per unit, and the top-of-the-line widget for $\$ 15$ per unit. The assembly line has a daily capacity of producing 350 widgets that may be sold for a total of $\$ 4600$. Find the quantity of each type of widget produced on a day when twice as many top-of-the-line widgets as basic widgets are produced.

Jim Long

Numerade Educator

Problem 99

Gary and Ginger decide to place $\$ 10,000$ of their savings into investments. They put some in a money market account earning $3 \%$ interest, some in a mutual fund that has been averaging $7 \%$ a year, and some in a stock that rose $10 \%$ last year. If they put $\$ 3000$ more in the money market than in the mutual fund and the mutual fund and stocks have the same growth in the next year as they did in the previous year, they will earn $\$ 540$ in a year. How much money did they put in each of the three investments?

James Kiss

Numerade Educator

Problem 100

Ginger talks Gary into putting less money in the money market and more money in the stock. They place $\$ 10,000$ of their savings into investments. They put some in a money market account earning $3 \%$ interest, some in a mutual fund that has been averaging $7 \%$ a year, and some in a stock that rose $10 \%$ last year. If they put $\$ 3000$ more in the stock than in the mutual fund and the mutual fund and stock have the same growth in the next year as they did in the previous year, they will earn $\$ 840$ in a year. How much money did they put in each of the three investments?

James Kiss

Numerade Educator

Problem 101

A company produces three products $x$, $y,$ and $z .$ Each item of product $x$ requires 20 units of steel, 2 units of plastic, and 1 unit of glass. Each item of product $y$ requires 25 units of steel, 5 units of plastic, and no units of glass. Each item of product $z$ requires 150 units of steel, 10 units of plastic, and 0.5 units of glass. The available amounts of steel, plastic, and glass are $2400,310,$ and 28 , respectively. How many items of each type can the company produce and utilize all the available raw materials?

James Kiss

Numerade Educator

Problem 102

Find the values of $a, b$, and $c$ such that the graph of the quadratic function $y=a x^{2}+b x+c$ passes through the points $(1,5),(-2,-10),$ and (0,4)

James Kiss

Numerade Educator

Problem 103

One hundred students decide to buy tickets to a football game. There are three types of tickets: general admission, reserved, and end zone. Each general admission ticket costs $\$ 20,$ each reserved ticket costs $\$ 40,$ and each end zone ticket costs $\$ 15 .$ The students spend a total of $\$ 2375$ for all the tickets. There are 5 more reserved tickets than general admission tickets and 20 more end zone tickets than general admission tickets. How many of each type of ticket were purchased by the students?

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 104

Ann would like to exercise one hour per day to burn calories and lose weight. She would like to engage in three activities: walking, step-up exercise, and weight training. She knows she can burn 85 calories walking at a certain pace in 15 minutes, 45 calories doing the step-up exercise in 10 minutes, and 137 calories by weight training for 20 minutes. (a) Determine the number of calories per minute she can burn doing each activity. (b) Suppose she has time to exercise for only one hour ( 60 minutes). She sets a goal of burning 358 calories in one hour and would like to weight train twice as long as walking. How many minutes must she engage in each exercise to burn the required number of calories in one hour?

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 105

The circle given by the equation $x^{2}+y^{2}+a x+b y+c=0$ passes through the point $(4,4),(-3,-1),$ and $(1,-3) .$ Find $a, b,$ and $c .$

James Kiss

Numerade Educator

Problem 106

The circle given by the equation $x^{2}+y^{2}+a x+b y+c=0$ passes through the point $(0,7),(6,1),$ and $(5,4) .$ Find $a, b,$ and $c .$

Ashley Volpe

Numerade Educator

Problem 107

In Exercises $107-110$, explain the mistake that is made.
Solve the system of equations using the augmented matrices
$$
\begin{array}{lr}
y-x+z= & 2 \\
x-2 z+y= & -3 \\
x+y+z= & 6
\end{array}
$$
Solution:
Step 1: Write as an augmented matrix.
$$
\left[\begin{array}{rrr|r}
1 & -1 & 1 & 2 \\
1 & -2 & 1 & -3 \\
1 & 1 & 1 & 6
\end{array}\right]
$$
Step 2: Reduce the matrix using Gaussian elimination.
$$
\left[\begin{array}{rrr|r}
1 & -1 & 1 & 2 \\
0 & 1 & 0 & 5 \\
0 & 0 & 0 & -6
\end{array}\right]
$$
Step
3: Identify the solution. Row 3 is inconsistent, so there is no solution.

This is incorrect. The correct answer is $x=1, y=2, z=3$. What mistake was made?

James Kiss

Numerade Educator

Problem 108

Explain the mistake that is made.
Perform the indicated row operations on the matrix.
$$
\left[\begin{array}{rrr|r}
1 & -1 & 1 & 2 \\
2 & -3 & 1 & 4 \\
3 & 1 & 2 & -6
\end{array}\right]
$$
a. $R_{2}-2 R_{1} \rightarrow R_{2}$
b. $R_{3}-3 R_{1} \rightarrow R_{3}$
Solution:
a. $\left[\begin{array}{rrr|r}1 & -1 & 1 & 2 \\ 0 & -3 & 1 & 4 \\ 3 & 1 & 2 & -6\end{array}\right]$
b. $\left[\begin{array}{rrr|r}1 & -1 & 1 & 2 \\ 2 & -3 & 1 & 4 \\ 0 & 1 & 2 & -6\end{array}\right]$
This is incorrect. What mistake was made?

James Kiss

Numerade Educator

Problem 109

Solve the system of equations using an augmented matrix.
$$
\begin{array}{rr}
3 x-2 y+z= & -1 \\
x+y-z= & 3 \\
2 x-y+3 z= & 0
\end{array}
$$
Solution:
Step 1: Write the system as an augmented matrix.
$$
\left[\begin{array}{rrr|r}
3 & -2 & 1 & -1 \\
1 & 1 & -1 & 3 \\
2 & -1 & 3 & 0
\end{array}\right]
$$
Step 2: Reduce the matrix using Gaussian elimination.
$$
\left[\begin{array}{lll|l}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & 2 \\
0 & 0 & 1 & 0
\end{array}\right]
$$
Step 3: Identify the answer. Row 3 is inconsistent $1=0$, therefore there is no solution.
This is incorrect. What mistake was made?

James Kiss

Numerade Educator

Problem 110

Solve the system of equations using an augmented matrix.
$$
\begin{array}{r}
x+3 y+2 z=4 \\
3 x+10 y+9 z=17 \\
2 x+7 y+7 z=17
\end{array}
$$
Solution:
Step 1: Write the system as an augmented matrix.
$$
\left[\begin{array}{rrr|r}
1 & 3 & 2 & 4 \\
3 & 10 & 9 & 17 \\
2 & 7 & 7 & 17
\end{array}\right]
$$
Step 2: Reduce the matrix using Gaussian elimination.
$$
\left[\begin{array}{rrr|r}
1 & 0 & -7 & -11 \\
0 & 1 & 3 & 5 \\
0 & 0 & 0 & 4
\end{array}\right]
$$
Step 3: Identify the answer:
$x=7 t-11$
Infinitely many solutions. $\quad y=-3 t+5$
$$
z=t
$$
This is incorrect. What mistake was made?

James Kiss

Numerade Educator

Problem 111

A nonsquare matrix cannot have a unique solution.

James Kiss

Numerade Educator

Problem 112

The procedure for Gaussian elimination can be used only for square matrices.

James Kiss

Numerade Educator

Problem 113

A square matrix that has a unique solution has a reduced matrix with 1 s along the diagonal and 0 s above and below the $1 \mathrm{~s}$.

James Kiss

Numerade Educator

Problem 114

A square matrix with an all-zero row has infinitely many solutions.

James Kiss

Numerade Educator

Problem 115

A fourth-degree polynomial $f(x)=a x^{4}+b x^{3}+c x^{2}+d x+e$, with $a<0,$ can be used to represent the following data on the number of deaths per year due to lightning strikes (assume 2012 corresponds to $x=0$ ).
Use the data to determine $a, b, c, d,$ and $e$.

Julie Silva

Numerade Educator

Problem 116

A copy machine accepts nickels, dimes, and quarters. After 1 hour, there are 30 coins total, and their value is $\$ 4.60 .$ How many nickels, quarters, and dimes are in the machine?

Erik Keohane

Numerade Educator

Problem 117

In Exercise $57,$ you were asked to solve this system of equations using an augmented matrix.
$$
\begin{array}{rr}
x-z-y= & 10 \\
2 x-3 y+z= & -11 \\
y-x+z= & -10
\end{array}
$$
A graphing calculator or graphing utility can be used to solve systems of linear equations by entering the coefficients of the matrix. Solve this system and confirm your answer with the calculator's answer.

James Kiss

Numerade Educator

Problem 118

In Exercise $58,$ you were asked to solve this system of equations using an augmented matrix.
$$
\begin{aligned}
2 x+z+y &=-3 \\
2 y-z+x &=0 \\
x+y+2 z &=5
\end{aligned}
$$
A graphing calculator or graphing utility can be used to solve systems of linear equations by entering the coefficients of the matrix. Solve this system and confirm your answer with the calculator's answer.

James Kiss

Numerade Educator

Problem 119

In Exercises 119 and $120,$ you are asked to model a set of three points with a quadratic function $y=a x^{2}+b x+c$ and determine the quadratic function.
a. Set up a system of equations and use a graphing utility or graphing calculator to solve the system by entering the coefficients of the augmented matrix.
b. Use the graphing calculator commands STAT QuadReg to model the data using a quadratic function. Round your answers to two decimal places.
$$
(-6,-8),(2,7),(7,1)
$$

James Kiss

Numerade Educator

Problem 120

You are asked to model a set of three points with a quadratic function $y=a x^{2}+b x+c$ and determine the quadratic function.
a. Set up a system of equations and use a graphing utility or graphing calculator to solve the system by entering the coefficients of the augmented matrix.
b. Use the graphing calculator commands STAT QuadReg to model the data using a quadratic function. Round your answers to two decimal places.
$$
(-9,20),(2,-18),(11,16)
$$

James Kiss

Numerade Educator