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

Amrita B.

Numerade Educator

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

Hannah T.

Numerade Educator

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

Amrita B.

Numerade Educator

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

Hannah T.

Numerade Educator

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

Amrita B.

Numerade Educator

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

Hannah T.

Numerade Educator

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

Amrita B.

Numerade Educator

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.

Hannah T.

Numerade Educator

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

$ \left[\begin{array}{rr}

7 & & 0

\end{array}\right] $

Amrita B.

Numerade Educator

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

$ \left[\begin{array}{rrrr}

5 & & -3 & & 8 & & 7

\end{array}\right] $

Hannah T.

Numerade Educator

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

$ \left[\begin{array}{r}

2 \\

36 \\

3

\end{array}\right] $

Amrita B.

Numerade Educator

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] $

Hannah T.

Numerade Educator

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

$ \left[\begin{array}{rr}

33 & & 45\\

-9 & & 20

\end{array}\right] $

Amrita B.

Numerade Educator

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

$ \left[\begin{array}{rrr}

-7 & & 6 & & 4\\

0 & & -5 & & 1

\end{array}\right] $

Hannah T.

Numerade Educator

In Exercises 15-20, write the augmented matrix for the system of linear equations.

$

\left\{

\begin{array}{l}

4x - 3y = -5 \\

-x + 3y = 12

\end{array}

\right.

$

Amrita B.

Numerade Educator

In Exercises 15-20, write the augmented matrix for the system of linear equations.

$

\left\{

\begin{array}{l}

7x + 4y = 22 \\

5x - 9y = 15

\end{array}

\right.

$

Hannah T.

Numerade Educator

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

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

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

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

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

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

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

$ \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

$ \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

$ \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

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

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

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

$ \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

$ \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

$ \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

$ \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

$ \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

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

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

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

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

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 \\

\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 $R_1$.

Oswaldo J.

Numerade Educator

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 $R_1$ to $R_4$.

(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 $R_4$.

Hannah T.

Numerade Educator

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 & 0 \\

0 & 1 & 1 & 5 \\

0 & 0 & 0 & 0 \\

\end{array}\right] $

Amrita B.

Numerade Educator

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 & 3 & 0 & 0 \\

0 & 0 & 1 & 8 \\

0 & 0 & 0 & 0 \\

\end{array}\right] $

Hannah T.

Numerade Educator

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 \\

\end{array}\right] $

Amrita B.

Numerade Educator

$ \left[\begin{array}{rr}

1 & 0 & 1 & 0 \\

0 & 1 & 0 & 2 \\

0 & 0 & 1 & 0 \\

\end{array}\right] $

Hannah T.

Numerade Educator

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

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

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

$ \left[\begin{array}{rr}

1 & -3 & 0 & -7 \\

-3 & 10 & 1 & 23 \\

4 & -10 & 2 & -24 \\

\end{array}\right] $

Hannah T.

Numerade Educator

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 & -4 \\

\end{array}\right] $

Amrita B.

Numerade Educator

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 & 3 & 2 \\

5 & 15 & 9 \\

2 & 6 & 10 \\

\end{array}\right] $

Hannah T.

Numerade Educator

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

$ \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

$ \left[\begin{array}{rrr}

-3 && 5 && 1 && 12 \\

1 && -1 && 1 && 4 \\

\end{array}\right] $

Amrita B.

Numerade Educator

$ \left[\begin{array}{rrr}

5 && 1 && 2 && 4 \\

-1 && 5 && 10 && -32 \\

\end{array}\right] $

Hannah T.

Numerade Educator

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

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

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

$ \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

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

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

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

$ \left[\begin{array}{rrrrr}

1 && 0 && 0 && \vdots && 5 \\

0 && 1 && 0 && \vdots && -3 \\

0 && 0 && 1 && \vdots && 0 \\

\end{array}\right] $

Hannah T.

Numerade Educator

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 = 7 \\

2x + y = 8

\end{array}

\right.

$

Amrita B.

Numerade Educator

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 = 16 \\

2x + 3y = 7

\end{array}

\right.

$

Hannah T.

Numerade Educator

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}

3x - 2y = -27 \\

x + 3y = 13

\end{array}

\right.

$

Oswaldo J.

Numerade Educator

$

\left\{

\begin{array}{l}

-x + y = 4 \\

2x - 4y = -34

\end{array}

\right.

$

Hannah T.

Numerade Educator

$

\left\{

\begin{array}{l}

-2x + 6y = -22 \\

x + 2y = -9

\end{array}

\right.

$

Wendy T.

Numerade Educator

$

\left\{

\begin{array}{l}

5x - 5y = -5 \\

-2x - 3y = 7

\end{array}

\right.

$

Hannah T.

Numerade Educator

$

\left\{

\begin{array}{l}

8x - 4y = 7 \\

5x + 2y = 1

\end{array}

\right.

$

Amrita B.

Numerade Educator

$

\left\{

\begin{array}{l}

x - 3y = 5 \\

-2x + 6y = -10

\end{array}

\right.

$

Hannah T.

Numerade Educator

$

\left\{

\begin{array}{l}

x - 3z = -2 \\

3x + y - 2z = 5 \\

2x + 2y + z = 4

\end{array}

\right.

$

Supratim R.

Numerade Educator

$

\left\{

\begin{array}{l}

2x - y + 3z = 24 \\

2y - z = 14 \\

7x - 5y = 6

\end{array}

\right.

$

Hannah T.

Numerade Educator

$

\left\{

\begin{array}{l}

-x + y - z = -14 \\

2x - y + z = 21 \\

3x + 2y + z = 19

\end{array}

\right.

$

Oswaldo J.

Numerade Educator

$

\left\{

\begin{array}{l}

2x + 2y - z = 2 \\

x - 3y + z = -28 \\

-x + y = 14

\end{array}

\right.

$

Hannah T.

Numerade Educator

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

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

Oswaldo J.

Numerade Educator

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

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

Oswaldo J.

Numerade Educator

$

\left\{

\begin{array}{l}

x + 2y = 0 \\

-x - y = 0

\end{array}

\right.

$

Amrita B.

Numerade Educator

$

\left\{

\begin{array}{l}

x + 2y = 0 \\

2x + 4y = 0

\end{array}

\right.

$

Hannah T.

Numerade Educator

$

\left\{

\begin{array}{l}

x + 2y + z = 8 \\

3x + 7y + 6z = 26

\end{array}

\right.

$

Amrita B.

Numerade Educator

$

\left\{

\begin{array}{l}

x + y + 4z = 5 \\

2x + y - z = 9

\end{array}

\right.

$

Hannah T.

Numerade Educator

$

\left\{

\begin{array}{l}

-x + y = -22 \\

3x + 4y = 4 \\

4x - 8y = 32

\end{array}

\right.

$

Stuart W.

Numerade Educator

$

\left\{

\begin{array}{l}

x +2y = 0 \\

x + y = 6 \\

3x - 2y = 8

\end{array}

\right.

$

Hannah T.

Numerade Educator

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

$

\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

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

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

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

$

\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

$

\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

$

\left\{

\begin{array}{l}

x + 2y + z + 3w = 0 \\

x - y + w = 0 \\

y - z + 2w = 0 \\

\end{array}

\right.

$

Hannah T.

Numerade Educator

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

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

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

(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

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

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

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

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

Hannah T.

Numerade Educator

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.

$f(-1) = -5$

$f(1) = -1$

$f(2) = 1$

$f(3) = 11$

Oswaldo J.

Numerade Educator

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.

$f(-1) = 4$

$f(1) = 4$

$f(2) = 16$

$f(3) = 44$

Hannah T.

Numerade Educator

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.

$f(-2) = -7$

$f(-1) = 2$

$f(1) = -4$

$f(2) = -7$

Oswaldo J.

Numerade Educator

$f(-2) = -17$

$f(-1) = -5$

$f(1) = 1$

$f(2) = 7$

Hannah T.

Numerade Educator

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

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

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

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

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.

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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.

Hannah T.

Numerade Educator

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.

Numerade Educator

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

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

Oswaldo J.

Numerade Educator

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

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

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

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

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

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

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.

Numerade Educator

THINK ABOUT IT

(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.

Numerade Educator

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

Amrita B.

Numerade Educator

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.

Numerade Educator

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?

Amrita B.

Numerade Educator