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

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

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

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

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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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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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In Exercises 9–14, determine the order of the matrix.

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

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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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In Exercises 9–14, determine the order of the matrix.

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

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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.

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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.

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PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices.

$$\frac{4 x^{2}}{(x+1)^{2}(x-1)}=\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{(x+1)^{2}}$$

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PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices.

$$\frac{8 x^{2}}{(x-1)^{2}(x+1)}=\frac{A}{x+1}+\frac{B}{x-1}+\frac{C}{(x-1)^{2}}$$

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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.

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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.

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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.

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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.

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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

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GRAPH CANNOT COPY

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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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