Problem 1

A rectangular array of real numbers that can be used to solve a system of linear equations is called

a ______.

Check back soon!

Problem 3

For a square matrix, the entries $a_{11}, a_{22}, a_{33}, \dots$ are the ________ ________ entries.

Check back soon!

Problem 4

A matrix with only one row is called a _______ matrix, and a matrix with only one column is called a ______ matrix.

Check back soon!

Problem 5

The matrix derived from a system of linear equations is called the _________ matrix of the system.

Check back soon!

Problem 6

The matrix derived from the coefficients of a system of linear equations is called the _______ matrix

of the system.

Check back soon!

Problem 7

. Two matrices are called _______ when one of the matrices can be obtained from the other by a

sequence of elementary row operations.

Check back soon!

Problem 8

. A matrix in row-echelon form is in _____ _____ ____ when every column that has a leading 1

has zeros in every position above and below its leading 1.

Check back soon!

Problem 9

Order of a Matrix, determine the order of the matrix.

$$\left[ \begin{array}{ll}{7} & {0}\end{array}\right]$$

Check back soon!

Problem 10

Order of a Matrix, determine the order of the matrix.

$$\left[ \begin{array}{llll}{5} & {-3} & {8} & {7}\end{array}\right]$$

Check back soon!

Problem 11

Order of a Matrix, determine the order of the matrix.

$$\left[ \begin{array}{r}{2} \\ {36} \\ {3}\end{array}\right]$$

Check back soon!

Problem 12

Order of a Matrix, determine the order of the matrix.

$$\left[ \begin{array}{rrrr}{-3} & {7} & {15} & {0} \\ {0} & {0} & {3} & {3} \\ {1} & {1} & {6} &{7}\end{array}\right]$$

Check back soon!

Problem 13

Order of a Matrix, determine the order of the matrix.

$$\left[ \begin{array}{rr}{33} & {45} \\ {-9} & {20}\end{array}\right]$$

Check back soon!

Problem 14

Order of a Matrix, determine the order of the matrix.

$$\left[ \begin{array}{rrr}{-7} & {6} & {4} \\ {0} & {-5} & {1}\end{array}\right]$$

Check back soon!

Problem 15

Order of a Matrix, determine the order of the matrix.

$$\left[ \begin{array}{rrr}{1} & {6} & {-1} \\ {8} & {0} & {3} \\ {3} & {-9} & {9}\end{array}\right]$$

Check back soon!

Problem 16

Order of a Matrix, determine the order of the matrix.

$$\left[ \begin{array}{rr}{3} & {-1} \\ {4} & {1} \\ {-5} & {9}\end{array}\right]$$

Check back soon!

Problem 17

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.

$$\left\{\begin{array}{ccc}{4 x-3 y} & {=} & {-5} \\ {-x+3 y} & {=} & {12}\end{array}\right.$$

Check back soon!

Problem 18

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.

$$\left\{\begin{array}{l}{7 x+4 y=22} \\ {5 x-9 y=15}\end{array}\right.$$

Check back soon!

Problem 19

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.

$$\left\{\begin{array}{r}{x+10 y-2 z=2} \\ {5 x-3 y+4 z=0} \\ {2 x+y \quad=6}\end{array}\right.$$

Check back soon!

Problem 20

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.

$$\left\{\begin{array}{rr}{-x-8 y+5 z=} & {8} \\ {-7 x-15 z=} & {-38} \\ {3 x-y+8 z=} & {20}\end{array}\right.$$

Check back soon!

Problem 21

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.

$$\left\{\begin{aligned} 7 x-5 y+z &=13 \\ 19 x &-8 z=10 \end{aligned}\right.$$

Check back soon!

Problem 22

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.

$$\left\{\begin{aligned} 9 x+& 2 y-3 z=20 \\-25 y+11 z &=-5 \end{aligned}\right.$$

Check back soon!

Problem 23

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)

$$\left[ \begin{array}{rrrr}{1} & {2} & {\vdots} & {7} \\ {2} & {-3} & {\vdots} & {4}\end{array}\right]$$

Check back soon!

Problem 24

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)

$$\left[ \begin{array}{rrrr}{7} & {-5} & {\vdots} & {0} \\ {8} & {3} & {\vdots} & {-2}\end{array}\right]$$

Check back soon!

Problem 25

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)

$$\left[ \begin{array}{rrrr}{2} & {0} & {5} & {\vdots} & {-12} \\ {0} & {1} & {-2} & {\vdots} & {7} \\ {6} & {3} & {0} & {\vdots} & {2}\end{array}\right]$$

Check back soon!

Problem 26

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)

$$\left[ \begin{array}{rrrrr}{4} & {-5} & {-1} & {\vdots} & {18} \\ {-11} & {0} & {6} & {\vdots} & {25} \\ {3} & {8} & {0} & {\vdots} & {-29}\end{array}\right]$$

Check back soon!

Problem 27

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)

$$\left[ \begin{array}{rrrrrr}{9} & {12} & {3} & {0} & {\vdots} & {0} \\ {-2} & {18} & {5} & {2} & {\vdots} & {10} \\ {1} & {7} & {-8} & {0} & {\vdots} & {-4} \\ {3} & {0} & {2} & {0} & {\vdots} & {-10}\end{array}\right]

$$

Check back soon!

Problem 28

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)

$$\left[ \begin{array}{rrrrrr}{6} & {2} & {-1} & {-5} & {\vdots} & {-25} \\ {-1} & {0} & {7} & {3} & {\vdots} & {7} \\ {4} & {-1} & {-10} & {6} & {\vdots} & {23} \\ {0} & {8} & {1} & {-11} & {\vdots} & {-21}\end{array}\right]$$

Check back soon!

Problem 29

Identifying an Elementary Row Operation, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.

$\\$ Original Matrix $\quad$ $\quad$$\quad$$\quad$$\quad$$\quad$ New Row-Equivalent Matrix $\\$

$\left[ \begin{array}{rrr}{-2} & {5} & {1} \\ {3} & {-1} & {-8}\end{array}\right]$ $\quad$ $\quad$ $\quad$ $\left[ \begin{array}{rrr}{13} & {0} & {-39} \\ {3} & {-1} & {-8}\end{array}\right]$

Check back soon!

Problem 30

Identifying an Elementary Row Operation, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.

$\\$ Original Matrix $\quad$ $\quad$$\quad$$\quad$$\quad$$\quad$New Row-Equivalent Matrix $\\$

$\left[ \begin{array}{rrr}{3} & {-1} & {-4} \\ {-4} & {3} & {7}\end{array}\right]$$\quad$$\quad$$\quad$$\quad$$\left[ \begin{array}{rrr}{3} & {-1} & {-4} \\ {5} & {0} & {-5}\end{array}\right]$

Check back soon!

Problem 31

Identifying an Elementary Row Operation, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.

$\\$ Original Matrix $\quad$ $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$New Row-Equivalent Matrix $\\$ $\left[ \begin{array}{rrrr}{0} & {-1} & {-5} & {5} \\ {-1} & {3} & {-7} & {6} \\ {4} & {-5} & {1} & {3}\end{array}\right]$

$\quad$$\quad$$\quad$$\quad$$\quad$ $\left[ \begin{array}{rrrr}{-1} & {3} & {-7} & {6} \\ {0} & {-1} & {-5} & {5} \\ {0} & {7} & {-27} & {27}\end{array}\right]$

Check back soon!

Problem 32

Identifying an Elementary Row Operation, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.

$\\$ Original Matrix $\quad$ $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$New Row-Equivalent Matrix $\\$$\left[ \begin{array}{rrrr}{-1} & {-2} & {3} & {-2} \\ {2} & {-5} & {1} & {-7} \\ {5} & {4} & {-7} & {6}\end{array}\right]$$\quad$$\quad$$\quad$$\quad$$\quad$$\left[ \begin{array}{rrrr}{-1} & {-2} & {3} & {-2} \\ {0} & {-9} & {7} & {-11} \\ {0} & {-6} & {8} & {-4}\end{array}\right]$

Check back soon!

Problem 33

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a

row-equivalent matrix.

$$\left[ \begin{array}{rrr}{3} & {6} & {8} \\ {4} & {-3} & {6}\end{array}\right]$$

$$\left[ \begin{array}{rrr}{1} & {4} & {3} \\ {0} & {} & {-1}\end{array}\right]$$

Check back soon!

Problem 34

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a

row-equivalent matrix.

$$\left[ \begin{array}{rrr}{3} & {6} & {8} \\ {4} & {-3} & {6}\end{array}\right]$$

$$\left[ \begin{array}{rrr}{1} & {} & {\frac{8}{3}} \\ {4} & {-3} & {{6}}\end{array}\right]$$

Check back soon!

Problem 35

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a

row-equivalent matrix.

$$\left[ \begin{array}{rrr}{1} & {1} & {1} \\ {5} & {-2} & {4}\end{array}\right]$$

$$\left[ \begin{array}{rrr}{1} & {1} & {{1}} \\ {0} & {} & {-1}\end{array}\right]$$

Check back soon!

Problem 36

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a

row-equivalent matrix.

$$\left[ \begin{array}{rrr}{-3} & {3} & {12} \\ {18} & {-8} & {4}\end{array}\right]$$

$$\left[ \begin{array}{rrr}{1} & {-1} & {} \\ {18} & {-8} & {4}\end{array}\right]$$

Check back soon!

Problem 37

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a

row-equivalent matrix.

$$\left[ \begin{array}{rrrr}{1} & {5} & {4} & {-1} \\ {0} & {1} & {-2} & {2} \\ {0} & {0} & {1} &{-7}\end{array}\right]$$

$$\left[ \begin{array}{rrrr}{1} & {0} & {} & {} \\ {0} & {1} & {-2} & {2} \\ {0} & {0} & {1} &{-7}\end{array}\right]$$

Check back soon!

Problem 38

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a

row-equivalent matrix.

$$\left[ \begin{array}{rrrr}{1} & {0} & {6} & {1} \\ {0} & {-1} & {0} & {7} \\ {0} & {0} & {-1} &{3}\end{array}\right]$$

$$\left[ \begin{array}{rrrr}{1} & {0} & {6} & {1} \\ {0} & {1} & {0} & {} \\ {0} & {0} & {1} &{}\end{array}\right]$$

Check back soon!

Problem 39

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a

row-equivalent matrix.

$$\left[ \begin{array}{rrrr}{1} & {1} & {4} & {-1} \\ {3} & {8} & {10} & {3} \\ {-2} & {1} & {12} & {6}\end{array}\right]$$

$$\left[ \begin{array}{rrrr}{1} & {1} & {4} & {-1} \\ {0} & {5} & {} & {} \\ {0} & {3} & {} & {}\end{array}\right]$$

$$\left[ \begin{array}{rrrr}{1} & {1} & {4} & {-1} \\ {0} & {1} & {-\frac{2}{5}} & {\frac{6}{5}} \\ {0} & {3} & {} & {}\end{array}\right]$$

Check back soon!

Problem 40

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a

row-equivalent matrix.

$$\left[ \begin{array}{rrrr}{2} & {4} & {8} & {3} \\ {1} & {-1} & {-3} & {2} \\ {2} & {6} & {4} & {9}\end{array}\right]$$

$$\left[ \begin{array}{rrrr}{1} & {} & {} & {} \\ {1} & {-1} & {-3} & {2} \\ {2} & {6} & {4} & {9}\end{array}\right]$$

$$\left[ \begin{array}{rrrr}{1} & {2} & {4} & {\frac{3}{2}} \\ {0} & {} & {-7} & {\frac{1}{2}} \\ {0} & {2} & {} & {}\end{array}\right]$$

Check back soon!

Problem 41

Comparing Linear Systems and Matrix

Operations In Exercises 41 and $42,$ (a) perform the

row operations to solve the augmented matrix, (b) write

and solve the system of linear equations represented by

the augmented matrix, and (c) compare the two solution

methods. Which do you prefer?

$$\left[ \begin{array}{rrrr}{-3} & {4} & {\vdots} & {22} \\ {6} & {-4} & {\vdots} & {-28}\end{array}\right]$$

$$\begin{array}{l}{\text { (i) Add } R_{2} \text { to } R_{1} \text { . }} \\ {\text { (ii) Add }-2 \text { times } R_{1} \text { to } R_{2} \text { . }} \\ {\text { (iii) Multiply } R_{2} \text { by }-\frac{1}{4}} \\ {\text { (iv) Multiply } R_{1} \text { by } \frac{1}{3}}\end{array}$$

Check back soon!

Problem 42

Comparing Linear Systems and Matrix

Operations In Exercises 41 and $42,$ (a) perform the

row operations to solve the augmented matrix, (b) write

and solve the system of linear equations represented by

the augmented matrix, and (c) compare the two solution

methods. Which do you prefer?

$$\left[ \begin{array}{rrrr}{7} & {13} & {1} & {\vdots} & {-4} \\ {-3} & {-5} & {-1} & {\vdots} & {-4} \\ {3} & {6} & {1} & {\vdots} & {-2}\end{array}\right]$$

$$\begin{array}{l}{\text { (i) Add } R_{2} \text { to } R_{1} \text { . }} \\ {\text { (ii) Multiply } R_{1} \text { by } \frac{1}{4}} \\ {\text { (iii) } \text { Add } R_{3} \text { to } R_{2} \text { . }}\end{array}$$

$$\begin{array}{l}{\text { (iv) } \mathrm{Add}-3 \text { times } R_{1} \text { to } R_{3} \text { . }} \\ {\text { (v) } \mathrm{Add}-2 \text { times } R_{2} \text { to } R_{1} .}\end{array}$$

Check back soon!

Problem 43

Row-Echelon Form, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.

$$\left[ \begin{array}{llll}{1} & {0} & {0} & {0} \\ {0} & {1} & {1} & {5} \\ {0} & {0} & {0} & {0}\end{array}\right]$$

Check back soon!

Problem 44

Row-Echelon Form, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.

$$\left[ \begin{array}{llll}{1} & {3} & {0} & {0} \\ {0} & {0} & {1} & {8} \\ {0} & {0} & {0} & {0}\end{array}\right]$$

Check back soon!

Problem 45

Row-Echelon Form, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.

$$\left[ \begin{array}{rrrr}{1} & {0} & {0} & {1} \\ {0} & {1} & {0} & {-1} \\ {0} & {0} & {0} & {2}\end{array}\right]$$

Check back soon!

Problem 46

Row-Echelon Form, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.

$$\left[ \begin{array}{llll}{1} & {0} & {1} & {0} \\ {0} & {1} & {0} & {2} \\ {0} & {0} & {1} & {0}\end{array}\right]$$

Check back soon!

Problem 47

Writing a Matrix in Row-Echelon Form, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)

$$\left[ \begin{array}{rrrr}{1} & {1} & {0} & {5} \\ {-2} & {-1} & {2} & {-10} \\ {3} & {6} & {7} & {14}\end{array}\right]$$

Check back soon!

Problem 48

Writing a Matrix in Row-Echelon Form, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)

$$\left[ \begin{array}{rrrr}{1} & {2} & {-1} & {3} \\ {3} & {7} & {-5} & {14} \\ {-2} & {-1} & {-3} & {8}\end{array}\right]$$

Check back soon!

Problem 49

Writing a Matrix in Row-Echelon Form, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)

$$\left[ \begin{array}{rrrr}{1} & {-1} & {-1} & {1} \\ {5} & {-4} & {1} & {8} \\ {-6} & {8} & {18} & {0}\end{array}\right]$$

Check back soon!

Problem 50

Writing a Matrix in Row-Echelon Form, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)

$$\left[ \begin{array}{rrrr}{1} & {-3} & {0} & {-7} \\ {-3} & {10} & {1} & {23} \\ {4} & {-10} & {2} & {-24}\end{array}\right]$$

Check back soon!

Problem 51

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

$$\left[ \begin{array}{rrr}{3} & {3} & {3} \\ {-1} & {0} & {-4} \\ {2} & {4} & {-2}\end{array}\right]$$

Check back soon!

Problem 52

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

$$\left[ \begin{array}{ccc}{1} & {3} & {2} \\ {5} & {15} & {9} \\ {2} & {6} & {10}\end{array}\right]$$

Check back soon!

Problem 53

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

$$\left[ \begin{array}{rrrr}{1} & {2} & {3} & {-5} \\ {1} & {2} & {4} & {-9} \\ {-2} & {-4} & {-4} & {3} \\ {4} & {8} & {11} & {-14}\end{array}\right]$$

Check back soon!

Problem 54

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

$$\left[ \begin{array}{rrrr}{-2} & {3} & {-1} & {-2} \\ {4} & {-2} & {5} & {8} \\ {1} & {5} & {-2} & {0} \\ {3} & {8} & {-10} & {-30}\end{array}\right]$$

Check back soon!

Problem 55

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

$$\left[ \begin{array}{rrrr}{-3} & {5} & {1} & {12} \\ {1} & {-1} & {1} & {4}\end{array}\right]$$

Check back soon!

Problem 56

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

$$\left[ \begin{array}{rrr}{5} & {1} & {2} & {4} \\ {-1} & {5} & {10} & {-32}\end{array}\right]$$

Check back soon!

Problem 57

Using Back-Substitution, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables $x, y,$ and $z,$ if applicable.)

$$\left[ \begin{array}{rrrr}{1} & {-2} & {\vdots} & {4} \\ {0} & {1} & {\vdots} & {-3}\end{array}\right]$$

Check back soon!

Problem 58

Using Back-Substitution, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables $x, y,$ and $z,$ if applicable.)

$$\left[ \begin{array}{llll}{1} & {5} & {\vdots} & {0} \\ {0} & {1} & {\vdots} & {-1}\end{array}\right]$$

Check back soon!

Problem 59

Using Back-Substitution, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables $x, y,$ and $z,$ if applicable.)

$$\left[ \begin{array}{rrrrr}{1} & {-1} & {2} & {\vdots} & {4} \\ {0} & {1} & {-1} & {\vdots} & {2} \\ {0} & {0} & {1} & {\vdots} & {-2}\end{array}\right]$$

Check back soon!

Problem 60

Using Back-Substitution, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables $x, y,$ and $z,$ if applicable.)

$$\left[ \begin{array}{rrrrr}{1} & {2} & {-2} & {\vdots} & {-1} \\ {0} & {1} & {1} & {\vdots} & {9} \\ {0} & {0} & {1} & {\vdots} & {-3}\end{array}\right]$$

Check back soon!

Problem 61

Interpreting Reduced Row-Echelon Form , an augmented matrix that represents a system of linear equations (in variables $x, y,$ and $z,$ if applicable) has been reduced using Gauss-Jordan

elimination. Write the solution represented by the augmented matrix.

$$\left\{\begin{aligned} 3 x+3 y+12 z &=6 \\ x+y+4 z &=2 \\ 2 x+5 y+20 z &=10 \\-x+2 y+8 z &=4 \end{aligned}\right.$$

Check back soon!

Problem 62

Interpreting Reduced Row-Echelon Form , an augmented matrix that represents a system of linear equations (in variables $x, y,$ and $z,$ if applicable) has been reduced using Gauss-Jordan

elimination. Write the solution represented by the augmented matrix.

$$\left[ \begin{array}{cccc}{1} & {0} & {\vdots} & {-6} \\ {0} & {1} & {\vdots} & {10}\end{array}\right]$$

Check back soon!

Problem 63

Interpreting Reduced Row-Echelon Form , an augmented matrix that represents a system of linear equations (in variables $x, y,$ and $z,$ if applicable) has been reduced using Gauss-Jordan

elimination. Write the solution represented by the augmented matrix.

$$\left[ \begin{array}{rrrr}{1} & {0} & {0} & {\vdots} & {-4} \\ {0} & {1} & {0} & {\vdots} & {-10} \\ {0} & {0} & {1} & {\vdots} & {4}\end{array}\right]$$

Check back soon!

Problem 64

Interpreting Reduced Row-Echelon Form , an augmented matrix that represents a system of linear equations (in variables $x, y,$ and $z,$ if applicable) has been reduced using Gauss-Jordan

elimination. Write the solution represented by the augmented matrix.

$$\left[ \begin{array}{rrrrr}{1} & {0} & {0} & {\vdots} & {5} \\ {0} & {1} & {0} & {\vdots} & {-3} \\ {0} & {0} & {1} & {\vdots} & {0}\end{array}\right]$$

Check back soon!

Problem 65

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.

$$\left\{\begin{aligned} x+2 y &=7 \\ 2 x+y &=8 \end{aligned}\right.$$

Check back soon!

Problem 66

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.

$$\left\{\begin{array}{r}{2 x+6 y=16} \\ {2 x+3 y=7}\end{array}\right.$$

Check back soon!

Problem 67

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.

$$\left\{\begin{array}{r}{3 x-2 y=-27} \\ {x+3 y=13}\end{array}\right.$$

Check back soon!

Problem 68

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.

$$\left\{\begin{aligned}-x+y &=4 \\ 2 x-4 y &=-34 \end{aligned}\right.$$

Check back soon!

Problem 69

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.

$$\left\{\begin{array}{rr}{x+2 y-3 z=} & {-28} \\ {4 y+2 z=} & {0} \\ {-x+y-z=} & {-5}\end{array}\right.$$

Check back soon!

Problem 70

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.

$$\left\{\begin{aligned} 3 x-2 y+z=& 15 \\-x+y+2 z=&-10 \\ x-y-4 z=& 14 \end{aligned}\right.$$

Check back soon!

Problem 71

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.

$$\left\{\begin{array}{rr}{-x+y=} & {-22} \\ {3 x+4 y=} & {4} \\ {4 x-8 y=} & {32}\end{array}\right.$$

Check back soon!

Problem 72

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.

$$\left\{\begin{aligned} x+2 y &=0 \\ x+y &=6 \\ 3 x-2 y &=8 \end{aligned}\right.$$

Check back soon!

Problem 73

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.

$$\left\{\begin{aligned} 3 x+2 y-z+w &=0 \\ x-y+4 z+2 w &=25 \\-2 x+y+2 z-w &=2 \\ x+y+z+w &=6 \end{aligned}\right.$$

Check back soon!

Problem 74

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.

$$\left\{\begin{aligned} x-4 y+3 z-2 w=& 9 \\ 3 x-2 y+z-4 w=&-13 \\-4 x+3 y-2 z+w=&-4 \\-2 x+y-4 z+3 w=&-10 \end{aligned}\right.$$

Check back soon!

Problem 75

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.

$$\left\{\begin{aligned}-2 x+6 y &=-22 \\ x+2 y &=-9 \end{aligned}\right.$$

Check back soon!

Problem 76

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.

$$\left\{\begin{aligned} 5 x-5 y &=-5 \\-2 x-3 y &=7 \end{aligned}\right.$$

Check back soon!

Problem 77

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.

$$\left\{\begin{array}{l}{8 x-4 y=7} \\ {5 x+2 y=1}\end{array}\right.$$

Check back soon!

Problem 78

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.

$$\left\{\begin{array}{rr}{x-3 y=} & {5} \\ {-2 x+6 y=} & {-10}\end{array}\right.$$

Check back soon!

Problem 79

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.

$$\left\{\begin{array}{c}{x+2 y+z=8} \\ {3 x+7 y+6 z=26}\end{array}\right.$$

Check back soon!

Problem 80

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.

$$\left\{\begin{aligned} x+y+4 z &=5 \\ 2 x+y-z &=9 \end{aligned}\right.$$

Check back soon!

Problem 81

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.

$$\left\{\begin{aligned} x &-3 z=-2 \\ 3 x+y-2 z &=5 \\ 2 x+2 y+z &=4 \end{aligned}\right.$$

Check back soon!

Problem 82

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.

$$\left\{\begin{aligned} 2 x-y+3 z &=24 \\ 2 y-z &=14 \\ 7 x-5 y &=6 \end{aligned}\right.$$

Check back soon!

Problem 83

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.

$$\left\{\begin{array}{rr}{-x+y-z=} & {-14} \\ {2 x-y+z=} & {21} \\ {3 x+2 y+z=} & {19}\end{array}\right.$$

Check back soon!

Problem 84

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.

$$\left\{\begin{aligned} 2 x+2 y-z &=2 \\ x-3 y+z &=-28 \\-x+y &=14 \end{aligned}\right.$$

Check back soon!

Problem 85

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.

$$\left\{\begin{aligned} 3 x+3 y+12 z &=6 \\ x+y+4 z &=2 \\ 2 x+5 y+20 z &=10 \\-x+2 y+8 z &=4 \end{aligned}\right.$$

Check back soon!

Problem 86

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.

$$\left\{\begin{aligned} 2 x+10 y+2 z=& 6 \\ x+5 y+2 z=& 6 \\ x+5 y+z=& 3 \\-3 x-15 y-3 z=&-9 \end{aligned}\right.$$

Check back soon!

Problem 87

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.

$$\left\{\begin{aligned} 2 x+y-z+2 w &=-6 \\ 3 x+4 y &+w=& 1 \\ x+5 y+2 z+6 w &=-3 \\ 5 x+2 y-z-w &=3 \end{aligned}\right.$$

Check back soon!

Problem 88

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.

$$\left\{\begin{aligned} x+2 y+2 z+4 w=& 11 \\ 3 x+6 y+5 z+12 w=& 30 \\ x+3 y-3 z+2 w=&-5 \\ 6 x-y-z+\quad w=&-9 \end{aligned}\right.$$

Check back soon!

Problem 89

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.

$$\left\{\begin{aligned} x+y+z+w &=0 \\ 2 x+3 y+z-2 w &=0 \\ 3 x+5 y+z &=0 \end{aligned}\right.$$

Check back soon!

Problem 90

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.

$$\left\{\begin{array}{r}{x+2 y+z+3 w=0} \\ {x-y+w=0} \\ {y-z+2 w=0}\end{array}\right.$$

Check back soon!

Problem 91

Comparing Solutions of Two Systems, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices.

$$(a)\left\{\begin{array}{rr}{x-2 y+z=} & {-6} \\ {y-5 z=} & {16} \\ {z=} & {-3}\end{array}\right.\quad(b)

\left\{\begin{aligned} x+y-2 z &=6 \\ y+3 z &=-8 \\ z &=-3 \end{aligned}\right.$$

Check back soon!

Problem 92

Comparing Solutions of Two Systems In, determine whether the two systems of

linear equations yield the same solution. If so, find the solution using matrices.

$$(a)\left\{\begin{array}{rr}{x-3 y+4 z=} & {-11} \\ {y-z=} & {-4} \\ {z=} & {2}\end{array}\right.$$

$$(b)\left\{\begin{array}{rr}{x+4 y} & {=-11} \\ {y+3 z} & {=4} \\ {z} & {=2}\end{array}\right.$$

Check back soon!

Problem 93

Comparing Solutions of Two Systems In, determine whether the two systems of

linear equations yield the same solution. If so, find the solution using matrices.

$$(a)\left\{\begin{array}{rr}{x-4 y+5 z=} & {27} \\ {y-7 z=} & {-54} \\ {z=} & {8}\end{array}\right.$$

$$(b)\left\{\begin{aligned} x-6 y+z &=15 \\ y+5 z &=42 \\ z &=8 \end{aligned}\right.$$

Check back soon!

Problem 94

Comparing Solutions of Two Systems In, determine whether the two systems of

linear equations yield the same solution. If so, find the solution using matrices.

$$(a)\left\{\begin{array}{rrr}{x+3 y-z=} & {19} \\ {y+6 z=} & {-18} \\ {z=} & {-4}\end{array}\right.$$

$$(b)\left\{\begin{array}{rrr}{x-y+3 z=} & {-15} \\ {y-2 z=} & {14} \\ {z=} & {-4}\end{array}\right.$$

Check back soon!

Problem 95

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.

$$f(1)=1, f(2)=-1, f(3)=-5$$

Check back soon!

Problem 96

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.

$$f(1)=2, f(2)=9, f(3)=20$$

Check back soon!

Problem 97

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.

$$f(-2)=-15, f(-1)=7, f(1)=-3$$

Check back soon!

Problem 98

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.

$$f(-2)=-3, f(1)=-3, f(2)=-11$$

Check back soon!

Problem 99

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.

$$f(1)=8, f(2)=13, f(3)=20$$

Check back soon!

Problem 100

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.

$$f(1)=9, f(2)=8, f(3)=5$$

Check back soon!

Problem 101

From 2000 through $2011,$ the numbers of new cases

of a waterborne disease in a small city increased in

a pattern that was approximately linear (see figure).

Find the least squares regression line

$$y=a t+b$$

for the data shown in the figure by solving the following system using matrices. Let $t$ represent the year, with $t=0$ corresponding to $2000 .$

$$\left\{\begin{aligned} 12 b+66 a &=831 \\ 66 b+506 a &=5643 \end{aligned}\right.$$

Use the result to predict the number of new cases

of the waterbome disease in $2014 .$ Is the estimate

reasonable? Explain.

Check back soon!

Problem 102

Breeding Facility A city zoo borrowed

$\$ 2,000,000$ at simple annual interest to construct a

breeding facility. Some of the money was borrowed at

$8 \%,$ some at $9 \%,$ and some at 12$\% .$ Use a system of

linear equations to determine how much was borrowed at each rate given that the total annual interest was

$\$ 186,000$ and the amount borrowed at 8$\%$ was twice

the amount borrowed at 12$\% .$ Solve the system of

linear equations using matrices.

Check back soon!

Problem 103

Museum A natural history museum borrowed

$\$ 2,000,000$ at simple annual interest to purchase new

exhibits. Some of the money was borrowed at $7 \%,$

some at $8.5 \%,$ and some at 9.5$\% .$ Use a system of

linear equations to determine how much was borrowed

at each rate given that the total annual interest was $\$ 169,750$ and the amount borrowed at 8.5$\%$ was four

times the amount borrowed at 9.5$\% .$ Solve the system

of linear equations using matrices.

Check back soon!

Problem 104

Mathematical Modeling A video of the path of

a ball thrown by a baseball player was analyzed with a

grid covering the TV screen. The tape was paused

three times, and the position of the ball was measured

each time. The coordinates obtained are shown in the

table. $(x$ and $y$ are measured in feet.)

$$\begin{array}{|c|c|c|c|c|}\hline \text { Horizontal Distance, } x & {0} & {15} & {30} \\ \hline \text { Height, y } & {5.0} & {9.6} & {12.4} \\ \hline\end{array}$$

$$\begin{array}{l}{\text { (a) Use a system of equations to find the equation of the }} \\ {\text { parabola } y=a x^{2}+b x+c \text { that passes through the }} \\ {\text { three points. Solve the system using matrices. }} \\ {\text { (b) Use a graphing utility to graph the parabola. }}\end{array}$$

$$\begin{array}{l}{\text { (c) Graphically approximate the maximum height of the }} \\ {\text { ball and the point at which the ball struck the ground. }} \\ {\text { (d) Analytically find the maximum height of the ball }} \\ {\text { and the point at which the ball struck the ground. }} \\ {\text { (e) Compare your results from parts (c) and (d). }}\end{array}$$

Check back soon!

Problem 105

True or False?, determine whether the statement is true or false. Justify your answer.

$\left[ \begin{array}{cccc}{5} & {0} & {-2} & {7} \\ {-1} & {3} & {-6} & {0}\end{array}\right]$ is a $4 \times 2$ matrix.

Check back soon!

Problem 106

The method of Gaussian elimination reduces a matrix

until a reduced row-echelon form is obtained.

Check back soon!

Problem 107

Think About It What is the relationship between

the three elementary row operations performed on an

augmented matrix and the operations that lead to

equivalent systems of equations?

Check back soon!

Problem 108

HOW DO YOU SEEIT? Determine whether

the matrix below is in row-echelon form,

reduced row-echelon form, or neither when

it satisfies the given conditions.

$$\left[ \begin{array}{ll}{1} & {b} \\ {c} & {1}\end{array}\right]$$

$$\begin{array}{ll}{\text { (a) } b=0, c=0} & {\text { (b) } b \neq 0, c=0} \\ {\text { (c) } b=0, c \neq 0} & {\text { (d) } b \neq 0, c \neq 0}\end{array}$$

Check back soon!