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 $a_{11}, a_{22}, a_{33}, \dots$ 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 _______ when 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 _____ _____ ____ when every column that has a leading 1

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

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

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

Kira H.

Numerade Educator

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.)

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

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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.)

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

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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.)

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

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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.)

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

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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]

$$

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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.)

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

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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 $\\$

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

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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]$

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

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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]$

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

row-equivalent matrix.

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

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

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

row-equivalent matrix.

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

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

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

row-equivalent matrix.

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

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

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

row-equivalent matrix.

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

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

Miranda H.

Numerade Educator

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

row-equivalent matrix.

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

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

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

row-equivalent matrix.

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

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

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

row-equivalent matrix.

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

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

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

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

row-equivalent matrix.

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

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

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

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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?

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

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

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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?

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

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

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

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

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

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

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

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

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

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

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

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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.)

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

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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.)

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

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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.)

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

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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.)

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.)

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

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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.)

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

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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.)

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

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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.)

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

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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.

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

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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.

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

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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.

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

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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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.

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

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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.

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

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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.

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

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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.

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

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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.

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

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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.

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

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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.$$

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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.

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

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

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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.

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

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

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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.

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

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

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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.

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

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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.

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

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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.

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

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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.

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

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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.

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

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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.

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

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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.

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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.

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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.

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

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

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

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

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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.

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The method of Gaussian elimination reduces a matrix

until a reduced row-echelon form is obtained.

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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?

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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.

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

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

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