# Precalculus with Limits (2010)

## Educators

Problem 1

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

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Problem 2

A matrix is ________ if the number of rows equals the number of columns.

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Problem 3

For a square matrix, the entries $a_{11}, a_{22}, a_{33}, \ldots, a_{n n}$ are the________,_______entries.

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Problem 4

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

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Problem 5

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

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Problem 6

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

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Problem 7

Two matrices are called ________ if one of the matrices can be obtained from the other by a sequence of elementary row operations.

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Problem 8

A matrix in row-echelon form is in ________ ________ ________ if every column that has a leading 1 has zeros in every position above and below its leading 1.

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Problem 9

In Exercises 9–14, determine the order of the matrix.
$$\left[\begin{array}{ll}{7} & {0}\end{array}\right]$$

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Problem 10

In Exercises 9–14, determine the order of the matrix.
$$\left[\begin{array}{llll}{5} & {-3} & {8} & {7}\end{array}\right]$$

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Problem 11

In Exercises 9–14, determine the order of the matrix.
$$\left[\begin{array}{r}{2} \\ {36} \\ {3}\end{array}\right]$$

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Problem 12

In Exercises 9–14, determine the order of the matrix.
$$\left[\begin{array}{rrrr}{-3} & {7} & {15} & {0} \\ {0} & {0} & {3} & {3} \\ {1} & {1} & {6} & {7}\end{array}\right]$$

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Problem 13

In Exercises 9–14, determine the order of the matrix.
$$\left[\begin{array}{cc}{33} & {45} \\ {-9} & {20}\end{array}\right]$$

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Problem 14

In Exercises 9–14, determine the order of the matrix.
$$\left[\begin{array}{rrr}{-7} & {6} & {4} \\ {0} & {-5} & {1}\end{array}\right]$$

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Problem 15

In Exercises 15–20, write the augmented matrix for the system of linear equations.
$$\left\{\begin{array}{l}{4 x-3 y=-5} \\ {-x+3 y=12}\end{array}\right.$$

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Problem 16

In Exercises 15–20, write the augmented matrix for the system of linear equations.
$$\left\{\begin{array}{l}{7 x+4 y=22} \\ {5 x-9 y=15}\end{array}\right.$$

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Problem 17

In Exercises 15–20, write the augmented matrix for the system of linear equations.
\left\{\begin{aligned} x+10 y-2 z &=2 \\ 5 x-3 y+4 z &=0 \\ 2 x+y &=6 \end{aligned}\right.

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Problem 18

In Exercises 15–20, write the augmented matrix for the system of linear equations.
$$\left\{\begin{array}{rr}{-x-8 y+5 z=} & {8} \\ {-7 x-15 z=}& {-38} \\ {3 x-y+8 z=} & {20}\end{array}\right.$$

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Problem 19

In Exercises 15–20, write the augmented matrix for the system of linear equations.
\left\{\begin{aligned} 7 x-5 y+z =13 \\ 19 x \quad -8 z=10 \end{aligned}\right.

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Problem 20

In Exercises 15–20, 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.

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Problem 21

In Exercises $21-26,$ write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z$ and $w,$ if applicable.)
$$\left[\begin{array}{rrrr}{1} & {2} & {\vdots} & {7} \\ {2} & {-3} & {\vdots} & {4}\end{array}\right]$$

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Problem 22

In Exercises $21-26,$ write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z$ and $w,$ if applicable.)
$$\left[\begin{array}{rrrr}{7} & {-5} & {\vdots} & {0} \\ {8} & {3} & {\vdots} & {-2}\end{array}\right]$$

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Problem 23

In Exercises $21-26,$ write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z$ and $w,$ if applicable.)
$$\left[\begin{array}{ccccc}{2} & {0} & {5} & {\vdots} & {-12} \\ {0} & {1} & {-2} & {\vdots} & {7} \\ {6} & {3} & {0} & {\vdots} & {2}\end{array}\right]$$

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Problem 24

In Exercises $21-26,$ write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z$ and $w,$ if applicable.)
$$\left[\begin{array}{rrrrr}{4} & {-5} & {-1} & {\vdots} & {18} \\ {-11} & {0} & {6} & {\vdots} & {25} \\ {3} & {8} & {0} & {\vdots} & {-29}\end{array}\right]$$

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Problem 25

In Exercises $21-26,$ write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z$ and $w,$ if applicable.)
$$\left[\begin{array}{rrrrrr}{9} & {12} & {3} & {0} & {\vdots} & {0} \\ {-2} & {18} & {5} & {2} & {\vdots} & {10} \\ {1} & {7} & {-8} & {0} & {\vdots} & {-4} \\ {3} & {0} & {2} & {0} & {\vdots} & {-10}\end{array}\right]$$

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Problem 26

In Exercises $21-26,$ write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z$ and $w,$ if applicable.)
$$\left[\begin{array}{rrrrrr}{6} & {2} & {-1} & {-5} & {\vdots} & {-25} \\ {-1} & {0} & {7} & {3} & {\vdots} & {7} \\ {4} & {-1} & {-10} & {6} & {\vdots} & {23} \\ {0} & {8} & {1} & {-11} & {\vdots} & {-21}\end{array}\right]$$

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Problem 27

In Exercises 27–34, fill in the blank(s) using elementary row operations to form a row-equivalent matrix.
$$\begin{array}{rrr}{\left[\begin{array}{ccc}{1} & {4} & {3} \\ {2} & {10} & {5}\end{array}\right]} \\ {\left[\begin{array}{rr}{1} & {4} & {3} \\ {0} & {} & {-1} \end{array}\right]}\end{array}$$

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Problem 28

In Exercises 27–34, fill in the blank(s) using elementary row operations to form a row-equivalent matrix.
$$\left[\begin{array}{rrr}{3} & {6} & {8} \\ {4} & {-3} & {6}\end{array}\right]\\\left[\begin{array}{rrr}{1} & {} & {\frac{8}{3}} \\ {4} & {-3} & {6}\end{array}\right]$$

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Problem 29

In Exercises 27–34, fill in the blank(s) using elementary row operations to form a row-equivalent matrix.
$$\left[\begin{array}{rrr}{1} & {1} & {1} \\ {5} & {-2} & {4}\end{array}\right]\\\left[\begin{array}{rr}{1} & {1} & {1} \\ {0} &{}& {-1}\end{array}\right]$$

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Problem 30

In Exercises 27–34, fill in the blank(s) using elementary row operations to form a row-equivalent matrix.
$$\left[\begin{array}{rrr}{-3} & {3} & {12} \\ {18} & {-8} & {4}\end{array}\right]\\\left[\begin{array}{rrr}{1} & {-1} & {} \\ {18} & {-8} & {4}\end{array}\right]$$

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Problem 31

In Exercises 27–34, fill in the blank(s) using elementary row operations to form a row-equivalent matrix.
$$\left[\begin{array}{rrrr}{1} & {5} & {4} & {-1} \\ {0} & {1} & {-2} & {2} \\ {0} & {0} & {1} & {-7}\end{array}\right]\\\left[\begin{array}{cccr}{1} & {0} \\ {0} & {1} & {-2} & {2} \\ {0} & {0} & {1} & {-7}\end{array}\right]$$

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Problem 32

In Exercises 27–34, fill in the blank(s) using elementary row operations to form a row-equivalent matrix.
$$\left[\begin{array}{rrrr}{1} & {0} & {6} & {1} \\ {0} & {-1} & {0} & {7} \\ {0} & {0} & {-1} & {3}\end{array}\right]\\\left[\begin{array}{cccc}{1} & {0} & {6} & {1} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right]$$

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Problem 33

In Exercises 27–34, fill in the blank(s) using elementary row operations to form a row-equivalent matrix.
$$\left[\begin{array}{rrrr}{1} & {1} & {4} & {-1} \\ {3} & {8} & {10} & {3} \\ {-2} & {1} & {12} & {6}\end{array}\right]\\\left[\begin{array}{cccc}{1} & {1} & {4} & {-1} \\ {0} & {5} & {} & {} \\ {0} & {3} & {} & {}\end{array}\right]\\\left[\begin{array}{rrrr}{1} & {1} & {4} & {-1} \\ {0} & {1} & {-\frac{2}{5}} & {\frac{6}{5}} \\ {0} & {3} & {}\end{array}\right]$$

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Problem 34

In Exercises 27–34, fill in the blank(s) using elementary row operations to form a row-equivalent matrix.
$$\left[\begin{array}{rrrr}{2} & {4} & {8} & {3} \\ {1} & {-1} & {-3} & {2} \\ {2} & {6} & {4} & {9}\end{array}\right]\\\left[\begin{array}{rrrr}{1} \\ {1} & {-1} & {-3} & {2} \\ {2} & {6} & {4} & {9}\end{array}\right]\\\left[\begin{array}{cccc}{1} & {2} & {4} & {\frac{3}{2}} \\ {0} & {} & {-7} & {\frac{1}{2}} \\ {0} & {2} & {}\end{array}\right]$$

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Problem 35

In Exercises 35–38, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.
$$\begin{array}{ll}{\text { Original Matrix }} & {\text { New Row-Equivalent Matrix }} \\ {\left[\begin{array}{rrr}{-2} & {5} & {1} \\ {3} & {-1} & {-8}\end{array}\right]} & {\left[\begin{array}{rrr}{13} & {0} & {-39} \\ {3} & {-1} & {-8}\end{array}\right]}\end{array}$$

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Problem 36

In Exercises 35–38, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.
$$\begin{array}{ll}{\text { Original Matrix }} & {\text { New Row-Equivalent Matrix }} \\ {\left[\begin{array}{rrr}{3} & {-1} & {-4} \\ {-4} & {3} & {7}\end{array}\right]} & {\left[\begin{array}{rrr}{3} & {-1} & {-4} \\ {5} & {0} & {-5}\end{array}\right]}\end{array}$$

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Problem 37

In Exercises 35–38, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.
$$\begin{array}{ll}{\text { Original Matrix }} & {\text { New Row-Equivalent Matrix }} \\ {\left[\begin{array}{rrr}{0} & {-1} & {-5} & {5} \\ {-1} & {3} & {-7} & {6} \\ {4} & {-5} & {1} & {3}\end{array}\right]} & {\left[\begin{array}{rrr}{-1} & {3} & {-7} & {6} \\ {0} & {-1} & {-5} & {5} \\ {0} & {7} & {-27} & {27}\end{array}\right]}\end{array}$$

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Problem 38

In Exercises 35–38, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.
$$\begin{array}{ll}{\text {Original Matrix}} & {\text {New Row-Equivalent Matrix}} \\ {\left[\begin{array}{rrrr}{-1} & {-2} & {3} & {-2} \\ {2} & {-5} & {1} & {-7} \\ {5} & {4} & {-7} & {6}\end{array}\right]} & {\left[\begin{array}{rrrr}{-1} & {-2} & {3} & {-2} \\ {0} & {-9} & {7} & {-11} \\ {0} & {-6} & {8} & {-4}\end{array}\right]}\end{array}$$

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Problem 39

Perform the sequence of row operations on the matrix. What did the operations accomplish?
$$\left[\begin{array}{rrr}{1} & {2} & {3} \\ {2} & {-1} & {-4} \\ {3} & {1} & {-1}\end{array}\right]$$
(a) Add $-2$ times $R_{1}$ to $R_{2}.$
(b) Add $-3$ times $R_{1}$ to $R_{3}.$
(c) Add $-1$ times $R_{2}$ to $R_{3}.$
(d) Multiply $R_{2}$ by $-\frac{1}{5.}$
(e) Add $-2$ times $R_{2}$ to $R_{1}.$

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Problem 40

Perform the sequence of row operations on the matrix. What did the operations accomplish?
$$\left[\begin{array}{rr}{7} & {1} \\ {0} & {2} \\ {-3} & {4} \\ {4} & {1}\end{array}\right]$$
(a) Add $R_{3}$ to $R_{4}.$
(b) Interchange $R_{1}$ and $R_{4}$ .
(c) Add 3 times $R_{1}$ to $R_{3}.$
(d) Add $-7$ times $R_{1}$ to $R_{4}.$
(e) Multiply $R_{2}$ by $\frac{1}{2}.$
(f) Add the appropriate multiples of $R_{2}$ to $R_{1}, R_{3},$ and $R_{4}$ .

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Problem 41

In Exercises 41–44, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form.
$$\left[\begin{array}{llll}{1} & {0} & {0} & {0} \\ {0} & {1} & {1} & {5} \\ {0} & {0} & {0} & {0}\end{array}\right]$$

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Problem 42

In Exercises 41–44, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form.
$$\left[\begin{array}{llll}{1} & {3} & {0} & {0} \\ {0} & {0} & {1} & {8} \\ {0} & {0} & {0} & {0}\end{array}\right]$$

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Problem 43

In Exercises 41–44, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form.
$$\left[\begin{array}{llll}{1} & {0} & {0} & {1} \\ {0} & {1} & {0} & {-1} \\ {0} & {0} & {0} & {2}\end{array}\right]$$

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Problem 44

In Exercises 41–44, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form.
$$\left[\begin{array}{llll}{1} & {0} & {1} & {0} \\ {0} & {1} & {0} & {2} \\ {0} & {0} & {1} & {0}\end{array}\right]$$

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Problem 45

In Exercises 45–48, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)
$$\left[\begin{array}{rrrr}{1} & {1} & {0} & {5} \\ {-2} & {-1} & {2} & {-10} \\ {3} & {6} & {7} & {14}\end{array}\right]$$

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Problem 46

In Exercises 45–48, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)
$$\left[\begin{array}{rrrr}{1} & {2} & {-1} & {3} \\ {3} & {7} & {-5} & {14} \\ {-2} & {-1} & {-3} & {8}\end{array}\right]$$

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Problem 47

In Exercises 45–48, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)
$$\left[\begin{array}{rrrr}{1} & {-1} & {-1} & {1} \\ {5} & {-4} & {1} & {8} \\ {-6} & {8} & {18} & {0}\end{array}\right]$$

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Problem 48

In Exercises 45–48, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)
$$\left[\begin{array}{rrrr}{1} & {-3} & {0} & {-7} \\ {-3} & {10} & {1} & {23} \\ {4} & {-10} & {2} & {-24}\end{array}\right]$$

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Problem 49

In Exercises 49–54, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[\begin{array}{rrr}{3} & {3} & {3} \\ {-1} & {0} & {-4} \\ {2} & {4} & {-2}\end{array}\right]$$

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Problem 50

In Exercises 49–54, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[\begin{array}{ccc}{1} & {3} & {2} \\ {5} & {15} & {9} \\ {2} & {6} & {10}\end{array}\right]$$

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Problem 51

In Exercises 49–54, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[\begin{array}{rrrr}{1} & {2} & {3} & {-5} \\ {1} & {2} & {4} & {-9} \\ {-2} & {-4} & {-4} & {3} \\ {4} & {8} & {11} & {-14}\end{array}\right]$$

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Problem 52

In Exercises 49–54, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[\begin{array}{rrrr}{-2} & {3} & {-1} & {-2} \\ {4} & {-2} & {5} & {8} \\ {1} & {5} & {-2} & {0} \\ {3} & {8} & {-10} & {-30}\end{array}\right]$$

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Problem 53

In Exercises 49–54, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[\begin{array}{rrrr}{-3} & {5} & {1} & {12} \\ {1} & {-1} & {1} & {4}\end{array}\right]$$

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Problem 54

In Exercises 49–54, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[\begin{array}{rrrr}{5} & {1} & {2} & {4} \\ {-1} & {5} & {10} & {-32}\end{array}\right]$$

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Problem 55

In Exercises $55-58$ , 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.)
$$\left[\begin{array}{rrrr}{1} & {-2} & {\vdots} & {4} \\ {0} & {1} & {\vdots} & {-3}\end{array}\right]$$

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Problem 56

In Exercises $55-58$ , 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.)
$$\left[\begin{array}{llll}{1} & {5} & {\vdots} & {0} \\ {0} & {1} & {\vdots} & {-1}\end{array}\right]$$

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Problem 57

In Exercises $55-58$ , 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.)
$$\left[\begin{array}{rrrrr}{1} & {-1} & {2} & {\vdots} & {4} \\ {0} & {1} & {-1} & {\vdots} & {2} \\ {0} & {0} & {1} & {\vdots} & {-2}\end{array}\right]$$

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Problem 58

In Exercises $55-58$ , 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.)
$$\left[\begin{array}{rrrrr}{1} & {2} & {-2} & {\vdots} & {-1} \\ {0} & {1} & {1} & {\vdots} & {9} \\ {0} & {0} & {1} & {\vdots} & {-3}\end{array}\right]$$

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Problem 59

In Exercises $59-62,$ 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.
$$\left[\begin{array}{llll}{1} & {0} & {\vdots} & {3} \\ {0} & {1} & {\vdots} & {-4}\end{array}\right]$$

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Problem 60

In Exercises $59-62,$ 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.
$$\left[\begin{array}{llll}{1} & {0} & {\vdots} & {-6} \\ {0} & {1} & {\vdots} & {10}\end{array}\right]$$

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Problem 61

In Exercises $59-62,$ 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.
$$\left[\begin{array}{ccccc}{1} & {0} & {0} & {\vdots} & {-4} \\ {0} & {1} & {0} & {\vdots} & {-10} \\ {0} & {0} & {1} & {\vdots} & {4}\end{array}\right]$$

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Problem 62

In Exercises $59-62,$ 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.
$$\left[\begin{array}{ccccc}{1} & {0} & {0} & {\vdots} & {5} \\ {0} & {1} & {0} & {\vdots} & {-3} \\ {0} & {0} & {1} & {\vdots} & {0}\end{array}\right]$$

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Problem 63

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} x+2 y &=7 \\ 2 x+y &=8 \end{aligned}\right.

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Problem 64

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{array}{l}{2 x+6 y=16} \\ {2 x+3 y=7}\end{array}\right.$$

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Problem 65

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} 3 x-2 y &=-27 \\ x+3 y &=13 \end{aligned}\right.

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Problem 66

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{array}{rr}{-x+y=} & {4} \\ {2 x-4 y=} & {-34}\end{array}\right.$$

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Problem 67

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned}-2 x+6 y &=-22 \\ x+2 y &=-9 \end{aligned}\right.

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Problem 68

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} 5 x-5 y &=-5 \\-2 x-3 y &=7 \end{aligned}\right.

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Problem 69

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{array}{l}{8 x-4 y=7} \\ {5 x+2 y=1}\end{array}\right.$$

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Problem 70

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} x-3 y &=5 \\-2 x+6 y &=-10 \end{aligned}\right.

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Problem 71

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or 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.

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Problem 72

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} 2 x-y+3 z &=24 \\ 2 y-z &=14 \\ {7 x-5 y}\qquad &=6 \end{aligned}\right.

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Problem 73

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned}-x+y-z=-14 \\ 2 x-y+z= 21 \\ 3 x+2 y+z= 19 \end{aligned}\right.

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Problem 74

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} 2 x+2 y-z &=2 \\ x-3 y+z &=-28 \\-x+y\qquad &=14 \end{aligned}\right.

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Problem 75

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} x+2 y-3 z=&-28 \\ 4 y+2 z=& 0 \\-x+y-z=&-5 \end{aligned}\right.

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Problem 76

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} 3 x-2 y+z=& 15 \\-x+y+2 z=&-10 \\ x-y-4 z=& 14 \end{aligned}\right.

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Problem 77

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} x+2 y &=0 \\-x-y &=0 \end{aligned}\right.

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Problem 78

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} x+2 y &=0 \\ 2 x+4 y &=0 \end{aligned}\right.

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Problem 79

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} x+2 y+z &=8 \\ 3 x+7 y+6 z &=26 \end{aligned}\right.

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Problem 80

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} x+y+4 z &=5 \\ 2 x+y-z &=9 \end{aligned}\right.

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Problem 81

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned}-x+y =-22 \\ 3 x+4 y =4 \\ 4 x-8 y =32 \end{aligned}\right.

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Problem 82

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\left\{\begin{aligned} x+2 y &=0 \\ x+y &=6 \\ 3 x-2 y &=8 \end{aligned}\right.

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Problem 83

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\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.

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Problem 84

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\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.

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Problem 85

In Exercises 85–90, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and 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.

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Problem 86

In Exercises 85–90, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system.
\left\{\begin{aligned} 2 x\quad+10 y+2 z= 6 \\ x\quad+5 y+2 z= 6 \\ x\qquad+5 y+z= 3 \\-3 x-15 y-3 z=-9 \end{aligned}\right.

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Problem 87

In Exercises 85–90, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system.
\left\{\begin{aligned} 2 x+y-z+2 w =-6 \\ 3 x+4 y \quad\quad\quad+w =1 \\ x+5 y+2 z+6 w =-3 \\ 5 x+2 y-z-w =3 \end{aligned}\right.

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Problem 88

In Exercises 85–90, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and 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+w &=-9 \end{aligned}\right.

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Problem 89

In Exercises 85–90, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system.
\left\{\begin{aligned} x+y+z+w &=0 \\ 2 x+3 y+z-2 w &=0\\ 3 x+5 y\quad\quad+z&=0 \end{aligned}\right.

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Problem 90

In Exercises 85–90, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system.
\left\{\begin{aligned} x+2 y+z+3 w =0 \\ x-y \qquad +w=0 \\ y-z+2 w =0 \end{aligned}\right.

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Problem 91

In Exercises 91–94, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices.
$$(a) \left\{\begin{array}{r}{x-2 y+z=-6} \\ {y-5 z=16} \\ {z=-3}\end{array}\right. \quad(b) \left\{\begin{array}{r}{x+y-2 z=6} \\ {y+3 z=-8} \\ {z=-3}\end{array}\right.$$

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Problem 92

In Exercises 91–94, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices.
$$(a) \left\{\begin{array}{r}{x-3 y+4 z=-11} \\ {y-z=-4} \\ {z=2}\end{array}\right. \quad(b) \left\{\begin{array}{r} {x+4 y =-11} \\ {y+3 z =4} \\ {z =2} \end{array}\right.$$

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Problem 93

In Exercises 91–94, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices.
$$(a)\left\{\begin{array}{r}{x-4 y+5 z=27} \\ {y-7 z=-54} \\ {z=8}\end{array}\right. \quad(b) \left\{\begin{array}{r}{x-6 y+z=15} \\ {y+5 z=42} \\ {z=8}\end{array}\right.$$

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Problem 94

In Exercises 91–94, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices.
$$(a)\left\{\begin{array}{r}{x+3 y-z=19} \\ {y+6 z=-18} \\ {z=-4}\end{array}\right.\quad(b) \left\{\begin{array}{r}{x-y+3 z=-15} \\ {y-2 z=14} \\ {z=-4}\end{array}\right.$$

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Problem 95

In Exercises $95-98,$ use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the equations. Solve the system using matrices.
$$f(1)=1, f(2)=-1, f(3)=-5$$

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Problem 96

In Exercises $95-98,$ use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the equations. Solve the system using matrices.
$$f(1)=2, f(2)=9, f(3)=20$$

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Problem 97

In Exercises $95-98,$ use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the equations. Solve the system using matrices.
$$f(-2)=-15, f(-1)=7, f(1)=-3$$

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Problem 98

In Exercises $95-98,$ use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the equations. Solve the system using matrices.
$$f(-2)=-3, f(1)=-3, f(2)=-11$$

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Problem 99

In Exercises $99-102,$ use a system of equations to find the cubic function $f(x)=a x^{3}+b x^{2}+c x+d$ that satisfies the equations. Solve the system using matrices.
$$\begin{array}{l}{f(-1)=-5} \\ {f(1)=-1} \\ {f(2)=1} \\ {f(3)=11}\end{array}$$

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Problem 100

In Exercises $99-102,$ use a system of equations to find the cubic function $f(x)=a x^{3}+b x^{2}+c x+d$ that satisfies the equations. Solve the system using matrices.
$$\begin{array}{l}{f(-1)=4} \\ {f(1)=4} \\ {f(2)=16} \\ {f(3)=44}\end{array}$$

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Problem 101

In Exercises $99-102,$ use a system of equations to find the cubic function $f(x)=a x^{3}+b x^{2}+c x+d$ that satisfies the equations. Solve the system using matrices.
$$\begin{array}{l}{f(-2)=-7} \\ {f(-1)=2} \\ {f(1)=-4} \\ {f(2)=-7}\end{array}$$

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Problem 102

In Exercises $99-102,$ use a system of equations to find the cubic function $f(x)=a x^{3}+b x^{2}+c x+d$ that satisfies the equations. Solve the system using matrices.
$$\begin{array}{l}{f(-2)=-17} \\ {f(-1)=-5} \\ {f(1)=1} \\ {f(2)=7}\end{array}$$

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Problem 103

Use the system
\left\{\begin{aligned} x+3 y+z &=3 \\ x+5 y+5 z &=1 \\ 2 x+6 y+3 z &=8 \end{aligned}\right. to write two different matrices in row-echelon form that yield the same solution. Check back soon! Problem 104 ELECTRICAL NETWORK The currents in an electrical network are given by the solution of the system\left\{\begin{aligned} I_{1}-I_{2}+I_{3} &=0 \\ 3 I_{1}+4 I_{2} &=18 \\ I_{2}+3 I_{3} &=6 \end{aligned}\right.$$where I_{1}, I_{2}, and I_{3} are measured in amperes. Solve the system of equations using matrices. Check back soon! Problem 105 PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices.$$\frac{4 x^{2}}{(x+1)^{2}(x-1)}=\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{(x+1)^{2}}$$Check back soon! Problem 106 PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices.$$\frac{8 x^{2}}{(x-1)^{2}(x+1)}=\frac{A}{x+1}+\frac{B}{x-1}+\frac{C}{(x-1)^{2}}$$Check back soon! Problem 107 FINANCE A small shoe corporation borrowed \ 1,500,000 to expand its line of shoes. Some of the money was borrowed at 7 \%, some at 8 \%, and some at 10\% . Use a system of equations to determine how much was borrowed at each rate if the annual interest was \ 130,500 and the amount borrowed at 10\% was 4 times the amount borrowed at 7\% . Solve the system using matrices. Check back soon! Problem 108 FINANCE A small software corporation borrowed \ 500,000 to expand its software line. Some of the money was borrowed at 9 \%, some at 10 \%, and some at 12\% . Use a system of equations to determine how much was borrowed at each rate if the annual interest was \ 52,000 and the amount borrowed at 10\% was 2\frac{1}{2} times the amount borrowed at 9\% . Solve the system using matrices. Check back soon! Problem 109 TIPS A food server examines the amount of money earned in tips after working an 8 -hour shift. The server has a total of \ 95 in denominations of \ 1, \ 5, \ 10, and \ 20 bills. The total number of paper bills is 26 . The number of \ 5 bills is 4 times the number of \ 10 bills, and the number of \ 1 bills is 1 less than twice the number of \ 5 bills. Write a system of linear equations to represent the situation. Then use matrices to find the number of each denomination. Check back soon! Problem 110 BANKING A bank teller is counting the total amount of money in a cash drawer at the end of a shift. There is a total of \ 2600 in denominations of \ 1, \ 5, \ 10 , and \ 20 bills The total number of paper bills is 235 . The number of \ 20 bills is twice the number of \ 1 bills, and the number of \ 5 bills is 10 more than the number of \ 1 bills. Write a system of linear equations to represent the situation. Then use matrices to find the number of each denomination. Check back soon! Problem 111 In Exercises 111 and 112, use a system of equations to find the equation of the parabola y=a x^{2}+b x+c that passes through the points. Solve the system using matrices. Use a graphing utility to verify your results. GRAPH CANNOT COPY Check back soon! Problem 112 In Exercises 111 and 112, use a system of equations to find the equation of the parabola y=a x^{2}+b x+c that passes through the points. Solve the system using matrices. Use a graphing utility to verify your results. GRAPH CANNOT COPY Check back soon! Problem 113 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 \text { and } y \text { are measured in feet.) }$$\begin{array}{|r|r|}\hline\text{Horizontal distance}&\text{x Height, y} \\{0} & {5.0} \\ {15} & {9.6} \\ {30} & {12.4}\\ \hline\end{array}

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