7 & & 0
\end{array}\right] $ Amrita B. Numerade Educator ### Problem 10 In Exercises 9-14, determine the order of the matrix.$ \left[\begin{array}{rrrr}
5 & & -3 & & 8 & & 7
2 \\
36 \\
3
\end{array}\right] $ Amrita B. Numerade Educator ### 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
33 & & 45\\
-9 & & 20
-7 & & 6 & & 4\\
0 & & -5 & & 1
\left\{
\begin{array}{l}
4x - 3y = -5 \\
-x + 3y = 12
\end{array}
\right.
\left\{
\begin{array}{l}
7x + 4y = 22 \\
5x - 9y = 15
\end{array}
\right.
$ Hannah T. Numerade Educator ### Problem 17 In Exercises 15-20, write the augmented matrix for the system of linear equations.$
\left\{
\begin{array}{l}
x + 10y - 2z = 2 \\
5x - 3y + 4z = 0 \\
2x + y = 6
\end{array}
\right.
$ Amrita B. Numerade Educator ### Problem 18 In Exercises 15-20, write the augmented matrix for the system of linear equations.$
\left\{
\begin{array}{l}
-x - 8y - 5z = 8 \\
-7x - 15z = -38 \\
3x - y + 8z = 20
\end{array}
\right.
$ Hannah T. Numerade Educator ### Problem 19 In Exercises 15-20, write the augmented matrix for the system of linear equations.$
\left\{
\begin{array}{l}
7x - 5y - z = 13 \\
19x - 8z = 10
\end{array}
\right.
$ Amrita B. Numerade Educator ### Problem 20 In Exercises 15-20, write the augmented matrix for the system of linear equations.$
\left\{
\begin{array}{l}
9x + 2y - 3z = 20 \\
-25y + 11z = -5
\end{array}
\right.
$ Hannah T. Numerade Educator ### 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] $ Amrita B. Numerade Educator ### 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] $ Hannah T. Numerade Educator ### 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}{rrrrr}
2 & 0 & & 5 & & \vdots & & -12 \\
0 & 1 & & -2 & & \vdots & & 7 \\
6 & 3 & & 0 & & \vdots & & 2
\end{array}\right] $ Amrita B. Numerade Educator ### 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] $ Hannah T. Numerade Educator ### 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] $ Amrita B. Numerade Educator ### 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] $ Hannah T. Numerade Educator ### Problem 27 In Exercises 27-34, fill in the blank(s) using elementary row operations to form a row-equivalent matrix.$ \left[\begin{array}{rrr}
1 & & 4 & & 3 \\
2 & & 10 & & 5 \\
\end{array}\right]  \left[\begin{array}{rrr}
1 & & 4 & & 3 \\
0 & & & & -1 \\
\end{array}\right] $ Mutahar M. Numerade Educator ### 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] $ Hannah T. Numerade Educator ### 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}{rrr}
1 & & 1 & & 1 \\
0 & & & & -1 \\
\end{array}\right] $ Amrita B. Numerade Educator ### 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] $ Hannah T. Numerade Educator ### 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}{rrrr}
1 & & 0 & & & & \\
0 & & 1 & & -2 & & 2 \\
0 & & 0 & & 1 & & -7 \\
\end{array}\right] $ Amrita B. Numerade Educator ### 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}{rrrr}
1 & & 0 & & 6 & & 1 \\
0 & & 1 & & 0 & & \\
0 & & 0 & & 1 & & \\
\end{array}\right] $ Hannah T. Numerade Educator ### 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}{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] $ Oswaldo J. Numerade Educator ### 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}{rrrr}
1 & & 2 & & 4 & & \frac{3}{2} \\
0 & & & & -7 & & \frac{1}{2} \\
0 & & 2 & & & & \\
\end{array}\right] $ Hannah T. Numerade Educator ### Problem 35 In Exercises 35-38, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. Original Matrix$ \left[\begin{array}{rrr}
-2 & & 5 & & 1 \\
3 & & -1 & & -8 \\
\end{array}\right] $New Row-Equivalent Matrix$ \left[\begin{array}{rrr}
13 & & 0 & & -39 \\
3 & & -1 & & -8 \\
\end{array}\right] $ Amrita B. Numerade Educator ### Problem 36 In Exercises 35-38, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. Original Matrix$ \left[\begin{array}{rrr}
3 & & -1 & & -4 \\
-4 & & 3 & & 7 \\
\end{array}\right] $New Row-Equivalent Matrix$ \left[\begin{array}{rrr}
3 & & -1 & & -4 \\
5 & & 0 & & -5 \\
\end{array}\right] $ Hannah T. Numerade Educator ### Problem 37 In Exercises 35-38, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. Original Matrix$ \left[\begin{array}{rrrr}
0 & & -1 & & -5 & & 5 \\
-1 & & 3 & & -7 & & 6 \\
4 & & -5 & & 1 & & 3 \\
\end{array}\right] $New Row-Equivalent Matrix$ \left[\begin{array}{rrrr}
-1 & & 3 & & -7 & & 6 \\
0 & & -1 & & -5 & & 5 \\
0 & & 7 & & -27 & & 27 \\
\end{array}\right] $ Amrita B. Numerade Educator ### Problem 38 In Exercises 35-38, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. Original Matrix$ \left[\begin{array}{rrrr}
-1 & & -2 & & 3 & & -2 \\
2 & & -5 & & 1 & & -7 \\
5 & & 4 & & -7 & & 6 \\
\end{array}\right] $New Row-Equivalent Matrix$ \left[\begin{array}{rrrr}
-1 & & -2 & & 3 & & -2 \\
0 & & -9 & & 7 & & -11 \\
0 & & -6 & & 8 & & -4 \\
\end{array}\right] $ Hannah T. Numerade Educator ### Problem 39 Perform the sequence of row operations on the matrix.What did the operations accomplish?$ \left[\begin{array}{rrrr}
1 & & 2 & & 3 \\
2 & & -1 & & -4 \\
3 & & 1 & & -1 \\
7 & 1 \\
0 & 2 \\
-3 & 4 \\
4 & 1\\
1 & 0 & 0 & 0 \\
0 & 1 & 1 & 5 \\
0 & 0 & 0 & 0 \\
1 & 3 & 0 & 0 \\
0 & 0 & 1 & 8 \\
0 & 0 & 0 & 0 \\
\end{array}\right] $ Hannah T. Numerade Educator ### 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}{rr}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & -1 \\
0 & 0 & 0 & 2 \\
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 2 \\
0 & 0 & 1 & 0 \\
\end{array}\right] $ Hannah T. Numerade Educator ### 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}{rr}
1 & 1 & 0 & 5 \\
-2 & -1 & 2 & -10 \\
3 & 6 & 7 & 14 \\
\end{array}\right] $ Oswaldo J. Numerade Educator ### 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}{rr}
1 & 2 & -1 & 3 \\
3 & 7 & -5 & 14 \\
-2 & -1 & -3 & 8 \\
\end{array}\right] $ Hannah T. Numerade Educator ### 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}{rr}
1 & -1 & -1 & 1 \\
5 & -4 & 1 & 8 \\
-6 & 8 & 18 & 0 \\
\end{array}\right] $ Oswaldo J. Numerade Educator ### 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}{rr}
1 & -3 & 0 & -7 \\
-3 & 10 & 1 & 23 \\
4 & -10 & 2 & -24 \\
3 & 3 & 3 \\
-1 & 0 & -4 \\
2 & 4 & -4 \\
1 & 3 & 2 \\
5 & 15 & 9 \\
2 & 6 & 10 \\
\end{array}\right] $ Hannah T. Numerade Educator ### 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}{rrr}
1 & 2 & 3 & -5 \\
1 & 2 & 4 & -9 \\
-2 & -4 & -4 & 3 \\
4 & 8 & 11 & -14 \\
\end{array}\right] $ Amrita B. Numerade Educator ### 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] $ Hannah T. Numerade Educator ### 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}{rrr}
-3 && 5 && 1 && 12 \\
1 && -1 && 1 && 4 \\
\end{array}\right] $ Amrita B. Numerade Educator ### 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}{rrr}
5 && 1 && 2 && 4 \\
-1 && 5 && 10 && -32 \\
\end{array}\right] $ Hannah T. Numerade Educator ### 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}{rrr}
1 && -2 && \vdots && 4 \\
0 && 1 && \vdots && -3 \\
\end{array}\right] $ Amrita B. Numerade Educator ### 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}{rrr}
1 && 5 && \vdots && 0 \\
0 && 1 && \vdots && -1 \\
\end{array}\right] $ Hannah T. Numerade Educator ### 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] $ Amrita B. Numerade Educator ### 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] $ Hannah T. Numerade Educator ### 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}{rrrrr}
1 && 0 && \vdots && 3 \\
0 && 1 && \vdots && -4 \\
\end{array}\right] $ Amrita B. Numerade Educator ### 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}{rrrrr}
1 && 0 && \vdots && -6 \\
0 && 1 && \vdots && 10 \\
\end{array}\right] $ Hannah T. Numerade Educator ### 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}{rrrrr}
1 && 0 && 0 && \vdots && -4 \\
0 && 1 && 0 && \vdots && -10 \\
0 && 0 && 1 && \vdots && 4 \\
\end{array}\right] $ Amrita B. Numerade Educator ### 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}{rrrrr}
1 && 0 && 0 && \vdots && 5 \\
0 && 1 && 0 && \vdots && -3 \\
0 && 0 && 1 && \vdots && 0 \\
\left\{
\begin{array}{l}
x + 2y = 7 \\
2x + y = 8
\end{array}
\right.
\left\{
\begin{array}{l}
2x + 6y = 16 \\
2x + 3y = 7
\end{array}
\right.
\left\{
\begin{array}{l}
3x - 2y = -27 \\
x + 3y = 13
\end{array}
\right.
\left\{
\begin{array}{l}
-x + y = 4 \\
2x - 4y = -34
\end{array}
\right.
$ Hannah T. Numerade Educator ### 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{array}{l}
-2x + 6y = -22 \\
x + 2y = -9
\end{array}
\right.
\left\{
\begin{array}{l}
5x - 5y = -5 \\
-2x - 3y = 7
\end{array}
\right.
\left\{
\begin{array}{l}
8x - 4y = 7 \\
5x + 2y = 1
\end{array}
\right.
\left\{
\begin{array}{l}
x - 3y = 5 \\
-2x + 6y = -10
\end{array}
\right.
$ Hannah T. Numerade Educator ### 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{array}{l}
x - 3z = -2 \\
3x + y - 2z = 5 \\
2x + 2y + z = 4
\end{array}
\right.
$ Supratim R. Numerade Educator ### 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{array}{l}
2x - y + 3z = 24 \\
2y - z = 14 \\
7x - 5y = 6
\end{array}
\right.
$ Hannah T. Numerade Educator ### 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{array}{l}
-x + y - z = -14 \\
2x - y + z = 21 \\
3x + 2y + z = 19
\end{array}
\right.
$ Oswaldo J. Numerade Educator ### 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{array}{l}
2x + 2y - z = 2 \\
x - 3y + z = -28 \\
-x + y = 14
\end{array}
\right.
\left\{
\begin{array}{l}
x + 2y = 0 \\
-x - y = 0
\end{array}
\right.
\left\{
\begin{array}{l}
x + 2y = 0 \\
2x + 4y = 0
\end{array}
\right.
$ Hannah T. Numerade Educator ### 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{array}{l}
x + 2y + z = 8 \\
3x + 7y + 6z = 26
\end{array}
\right.
$ Amrita B. Numerade Educator ### 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{array}{l}
x + y + 4z = 5 \\
2x + y - z = 9
\end{array}
\right.
\left\{
\begin{array}{l}
-x + y = -22 \\
3x + 4y = 4 \\
4x - 8y = 32
\end{array}
\right.
\left\{
\begin{array}{l}
x +2y = 0 \\
x + y = 6 \\
3x - 2y = 8
\end{array}
\right.
$ Hannah T. Numerade Educator ### Problem 83 Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$\left\{ \begin{array}{l} 3x + 2y - z + w = 0 \\ x - y + 4z + 2w = 25 \\ -2x + y + 2z + w = 2 \\ x + y + z + w = 6 \\ \end{array} \right.$$ Oswaldo J. Numerade Educator ### 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{array}{l}
x - 4y + 3z - 2w = 9 \\
3x - 2y + z - 4w = -13 \\
-4x + 3y - 2z + w = -4 \\
-2x + y - 4z +3w = -10 \\
\end{array}
\right.
$ Hannah T. Numerade Educator ### 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{array}{l}
3x + 3y + 12z = 6 \\
x + y + 4z = 2 \\
2x + 5y + 20z = 10 \\
-x + 2y + 8z = 4 \\
\end{array}
\right.
$ Oswaldo J. Numerade Educator ### 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{array}{l}
2x + 10y + 2z = 6 \\
x + 5y + 2z = 6 \\
x + 5y + z = 3 \\
-3x - 15y - 3z = -9 \\
\end{array}
\right.
$ Hannah T. Numerade Educator ### 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{array}{l}
2x + y - z + 2w = -6 \\
3x + 4y + w = 1 \\
x + 5y + 2z + 6w = -3 \\
5x + 2y - z + w = 3 \\
\end{array}
\right.
$ Oswaldo J. Numerade Educator ### 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{array}{l}
x + 2y + 2z + 4w = 11 \\
3x + 6y + 5z + 12w = 30 \\
x + 3y - 3z + 2w = -5 \\
6x - y - z + w = -9 \\
\end{array}
\right.
$ Hannah T. Numerade Educator ### 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{array}{l}
x + y + z + w = 0 \\
2x + 3y + z - 2w = 0 \\
3x + 5y + z = 0 \\
\end{array}
\right.
$ Oswaldo J. Numerade Educator ### 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{array}{l}
x + 2y + z + 3w = 0 \\
x - y + w = 0 \\
y - z + 2w = 0 \\
\end{array}
\right.
$ Hannah T. Numerade Educator ### 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}{l}
x - 2y + z = -6 \\
y - 5z = 16 \\
z = -3 \\
\end{array}
\right.
$(b)$
\left\{
\begin{array}{l}
x + y - 2z = 6 \\
y + 3z = -8 \\
z = -3 \\
\end{array}
\right.
$ Oswaldo J. Numerade Educator ### 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}{l}
x - 3y + 4z = -11 \\
y - z = -4 \\
z = 2 \\
\end{array}
\right.
$(b)$
\left\{
\begin{array}{l}
x + 4y = -11 \\
y + 3z = 4 \\
z = 2 \\
\end{array}
\right.
$ Hannah T. Numerade Educator ### 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}{l}
x - 4y + 5z = 27 \\
y - 7z = -54 \\
z = 8 \\
\end{array}
\right.
$(b)$
\left\{
\begin{array}{l}
x - 6y - z = 15 \\
y + 5z = 42 \\
z = 8 \\
\end{array}
\right.
$ Oswaldo J. Numerade Educator ### 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}{l}
x + 3y - z = 19 \\
y + 6z = -18 \\
z = -4 \\
\end{array}
\right.
$(b)$
\left\{
\begin{array}{l}
x - y + 3z = -15 \\
y - 2z = 14 \\
z = -4 \\
\end{array}
\right.
$ Hannah T. Numerade Educator ### Problem 95 In Exercises 95-98, use a system of equations to find the quadratic function$f(x) = ax^2 + bx + c$that satisfies the equations. Solve the system using matrices.$f(1) = 1$,$f(2) = -1$,$f(3) = -5$ Oswaldo J. Numerade Educator ### Problem 96 In Exercises 95-98, use a system of equations to find the quadratic function$f(x) = ax^2 + bx + c$that satisfies the equations. Solve the system using matrices.$f(1) = 2$,$f(2) = 9$,$f(3) = 20$ Hannah T. Numerade Educator ### Problem 97 In Exercises 95-98, use a system of equations to find the quadratic function$f(x) = ax^2 + bx + c$that satisfies the equations. Solve the system using matrices.$f(-2) = -15$,$f(-1) = 7$,$f(1) = -3$ Oswaldo J. Numerade Educator ### Problem 98 In Exercises 95-98, use a system of equations to find the quadratic function$f(x) = ax^2 + bx + c$that satisfies the equations. Solve the system using matrices.$f(-2) = -3$,$f(1) = -3$,$f(2) = -11$ Hannah T. Numerade Educator ### Problem 99 In Exercises 99-102, use a system of equations to find the cubic function$f(x) = ax^3 + bx^2 + cx + d$that satisfies the equations. Solve the system using matrices.$f(-1) = -5f(1) = -1f(2) = 1f(3) = 11$ Oswaldo J. Numerade Educator ### Problem 100 In Exercises 99-102, use a system of equations to find the cubic function$f(x) = ax^3 + bx^2 + cx + d$that satisfies the equations. Solve the system using matrices.$f(-1) = 4f(1) = 4f(2) = 16f(3) = 44$ Hannah T. Numerade Educator ### Problem 101 In Exercises 99-102, use a system of equations to find the cubic function$f(x) = ax^3 + bx^2 + cx + d$that satisfies the equations. Solve the system using matrices.$f(-2) = -7f(-1) = 2f(1) = -4f(2) = -7$ Oswaldo J. Numerade Educator ### Problem 102 In Exercises 99-102, use a system of equations to find the cubic function$f(x) = ax^3 + bx^2 + cx + d$that satisfies the equations. Solve the system using matrices.$f(-2) = -17f(-1) = -5f(1) = 1f(2) = 7$ Hannah T. Numerade Educator ### Problem 103 Use the system$
\left\{
\begin{array}{l}
x + 3y + z = 3 \\
x + 5y + 5z = 1 \\
2x + 6y + 3z = 8 \\
\end{array}
\right.
$to write two different matrices in row-echelon form that yield the same solution. Abdul V. Numerade Educator ### Problem 104 ELECTRICAL NETWORK The currents in an electrical network are given by the solution of the system$
\left\{
\begin{array}{l}
I_1 - I_2 + I_3 = 0 \\
3I_1 + 4I_2 = 18 \\
I_2 + 3I_3 = 6 \\
\end{array}
\right.
$where$I_1$,$I_2$, and$I_3$, are measured in amperes. Solve the system of equations using matrices. Hannah T. Numerade Educator ### Problem 105 PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices.$\dfrac{4x^2}{(x+1)^2(x-1)} = \dfrac{A}{x-1} + \dfrac{B}{x+1} + \dfrac{C}{(x+1)^2}$ Oswaldo J. Numerade Educator ### Problem 106 PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices.$\dfrac{8x^2}{(x-1)^2(x+1)} = \dfrac{A}{x+1} + \dfrac{B}{x-1} + \dfrac{C}{(x-1)^2}$ Hannah T. Numerade Educator ### 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 at10% 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. Hannah T. Numerade Educator ### 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. Jonathan S.

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 then umber of each denomination. Hannah T. Numerade Educator ### Problem 111 In Exercises 111 and 112, use a system of equations to find the equation of the parabola that passes through the points. Solve the system using matrices. Use a graphing utility to verify your results. Oswaldo J. Numerade Educator ### Problem 112 In Exercises 111 and 112, use a system of equations to find the equation of the parabola that passes through the points. Solve the system using matrices. Use a graphing utility to verify your results. Oswaldo J. Numerade Educator ### 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. ($x$and$y$are measured in feet.) (a) Use a system of equations to find the equation of the parabola$y = ax^2 + bx = c$that passes through the three points. Solve the system using matrices. (b) Use a graphing utility to graph the parabola. (c) Graphically approximate the maximum height of the ball and the point at which the ball struck the ground. (d) Analytically find the maximum height of the ball and the point at which the ball struck the ground. (e) Compare your results from parts (c) and (d). Oswaldo J. Numerade Educator ### Problem 114 DATA ANALYSIS: SNOW BOARDERS The tables hows the numbers of people$y$(in millions) in the United States who participated in snowboarding in selected years from 2003 to 2007. (Source: National Sporting Goods Association) (a) Use a system of equations to find the equation of the parabola$y = at^2 +bt + c$that passes through the points. Let$t$represent the year, with$t=3$corresponding to 2003. Solve the system using matrices. (b) Use a graphing utility to graph the parabola. (c) Use the equation in part (a) to estimate the number of people who participated in snowboarding in 2009. Does your answer seem reasonable? Explain. (d) Do you believe that the equation can be used for years far beyond 2007? Explain. Oswaldo J. Numerade Educator ### Problem 115 NETWORK ANALYSIS In Exercises 115 and 116, answer the questions about the specified network. (In a network it is assumed that the total flow into each junction is equal to the total flow out of each junction.) Water flowing through a network of pipes (in thousands of cubic meters per hour) is shown in the figure. (a) Solve this system using matrices for the water flow represented by$x_i$,$i = 1$,$2$,$\ldots$,$7$. (b) Find the network flow pattern when$x_6 = 0$and$x_7 = 0$. (c) Find the network flow pattern when$x_5 = 400$and$x_6 = 500$. Oswaldo J. Numerade Educator ### Problem 116 NETWORK ANALYSIS In Exercises 115 and 116, answer the questions about the specified network. (In a network it is assumed that the total flow into each junction is equal to the total flow out of each junction.) The flow of traffic (in vehicles per hour) through a network of streets is shown in the figure. (a) Solve this system using matrices for the traffic flow represented by$x_i$,$i = 1$,$2$,$\ldots$,$5$(b) Find the traffic flow when$x_2 = 200$and$x_3 = 50$. (c) Find the traffic flow when$x_2 = 150$and$x_3 = 0$. Oswaldo J. Numerade Educator ### Problem 117 TRUE OR FALSE? In Exercises 117 and 118, determine whether the statement is true or false. Justify your answer.$ \left[\begin{array}{rrrrr}
5 && 0 && -2 && 7 \\
-3 && 3 && -6 && 0 \\
\end{array}\right] $is 4 x 2 matrix. Amrita B. Numerade Educator ### Problem 118 TRUE OR FALSE? In Exercises 117 and 118, determine whether the statement is true or false. Justify your answer. The method of Gaussian elimination reduces a matrix until a reduced row-echelon form is obtained. Hannah T. Numerade Educator ### Problem 119 THINK ABOUT IT The augmented matrix below represents system of linear equations (in variables$x$,$y$, and$z$) that has been reduced using Gauss-Jordan elimination. Write a system of equations with nonzero coefficients that is represented by the reduced matrix. (There are many correct answers.)$ \left[\begin{array}{rrrrr}
1 && 0 && 3 && \vdots & -2 \\
0 && 1 && 4 && \vdots & 1 \\
0 && 0 && 0 && \vdots & 0 \\
\end{array}\right] \$ Amrita B.

### Problem 120

(a) Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that is inconsistent.
(b) Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has an infinite number of solutions. Hannah T.

### Problem 121

Describe the three elementary row operations that can be performed on an augmented matrix. Amrita B.

### Problem 122

CAPSTONE In your own words, describe the difference between a matrix in row-echelon form and a matrix in reduced row-echelon form. Include an example of each to support your explanation. Hannah T. 