Write the augmented matrix for each system of linear equations.

$\begin{cases} 2x + y + 2z = 2 \\ 3x - 5y - z = 4 \\ x - 2y - 3z = -6 \end{cases}$

Write the augmented matrix for each system of linear equations.

$\begin{cases} x - y + z = 8 \\ y - 12z = -15 \\ z = 1 \end{cases}$

Write the augmented matrix for each system of linear equations.

$\begin{cases} 5x - 2y - 3z = 0 \\ x + y = 5 \\ 2x - 3z = 4 \end{cases}$

Write the augmented matrix for each system of linear equations.

$\begin{cases} 2w + 5x - 3y + z = 2 \\ 3x + y = 4 \\ w - x + 5y = 9 \\ 5w - 5x - 2y = 1 \end{cases}$

Write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.

$\begin{bmatrix} 5 & 0 & 3 & \vert & -11 \\ 0 & 1 & -4 & \vert & 12 \\ 7 & 2 & 0 & \vert & 3 \end{bmatrix}$

Write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.

$\begin{bmatrix} 1 & 1 & 4 & 1 & \vert & 3 \\ -1 & 1 & -1 & 0 & \vert & 7 \\ 2 & 0 & 0 & 5 & \vert & 11 \\ 0 & 0 & 12 & 4 & \vert & 5 \end{bmatrix}$

Perform each matrix row operation and write the new matrix.

$\begin{bmatrix} 2 & -6 & 4 & \vert & 10 \\ 1 & 5 & -5 & \vert & 0 \\ 3 & 0 & 4 & \vert & 7 \end{bmatrix} \quad \frac{1}{2}R_1$

Perform each matrix row operation and write the new matrix.

$\begin{bmatrix} 1 & -3 & 2 & \vert & 0 \\ 3 & 1 & -1 & \vert & 7 \\ 2 & -2 & 1 & \vert & 3 \end{bmatrix} \quad -3R_1 + R_2$

Perform each matrix row operation and write the new matrix.

$\begin{bmatrix} 1 & -1 & 1 & 1 & \vert & 3 \\ 0 & 1 & -2 & -1 & \vert & 0 \\ 2 & 0 & 3 & 4 & \vert & 11 \\ 5 & 1 & 2 & 4 & \vert & 6 \end{bmatrix} \quad \begin{array}{l} -2R_1 + R_3 \\ -5R_1 + R_4 \end{array}$

In Exercises 19–20, a few steps in the process of simplifying the given matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s, are shown. Fill in the missing numbers in the steps that are shown.

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases} x + y - z = -2 \\ 2x - y + z = 5 \\ -x + 2y + 2z = 1 \end{cases}$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases} x + 3y = 0 \\ x + y + z = 1 \\ 3x - y - z = 11 \end{cases}$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases} 2x - y - z = 4 \\ x + y - 5z = -4 \\ x - 2y = 4 \end{cases}$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases} x + y + z = 4 \\ x - y - z = 0 \\ x - y + z = 2 \end{cases}$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases} x + 2y = z - 1 \\ x = 4 + y - z \\ x + y - 3z = -2 \end{cases}$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases} 3a - b - 4c = 3 \\ 2a - b + 2c = -8 \\ a + 2b - 3c = 9 \end{cases}$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases} 2x + 2y + 7z = -1 \\ 2x + y + 2z = 2 \\ 4x + 6y + z = 15 \end{cases}$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases} w + x + y + z = 4 \\ 2w + x - 2y - z = 0 \\ w - 2x - y - 2z = -2 \\ 3w + 2x + y + 3z = 4 \end{cases}$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases} 3w - 4x + y + z = 9 \\ w + x - y - z = 0 \\ 2w + x + 4y - 2z = 3 \\ -w + 2x + y - 3z = 3 \end{cases}$

Find the quadratic function f(x) = ax² + bx + c for which ƒ( − 2) = −4, ƒ(1) = 2, and f(2) = 0.

Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.

Solve the system: (Hint: Let A = ln w, B = ln x, C = ln y, and D = ln z. Solve the system for A, B, C, and D. Then use the logarithmic equations to find w, x, y, and z.)

$\begin{cases} 2 \ln w + \ln x + 3 \ln y - 2 \ln z = -6 \\ 4 \ln w + 3 \ln x + \ln y - \ln z = -2 \\ \ln w + \ln x + \ln y + \ln z = -5 \\ \ln w + \ln x - \ln y - \ln z = 5 \end{cases}$

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} 5x + 12y + z = 10 \\ 2x + 5y + 2z = -1 \\ x + 2y - 3z = 5 \end{cases}$

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} 5x + 8y - 6z = 14 \\ 3x + 4y - 2z = 8 \\ x + 2y - 2z = 3 \end{cases}$

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} 3x + 4y + 2z = 3 \\ 4x - 2y - 8z = -4 \\ x + y - z = 3 \end{cases}$

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} 8x + 5y + 11z = 30 \\ -x - 4y + 2z = 3 \\ 2x - y + 5z = 12 \end{cases}$

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} w - 2x - y - 3z = -9 \\ w + x - y = 0 \\ 3w + 4x + z = 6 \\ 2x - 2y + z = 3 \end{cases}$

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} 2w + x - y = 3 \\ w - 3x + 2y = -4 \\ 3w + x - 3y + z = 1 \\ w + 2x - 4y - z = -2 \end{cases}$

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} w - 3x + y - 4z = 4 \\ -2w + x + 2y = -2 \\ 3w - 2x + y - 6z = 2 \\ -w + 3x + 2y - z = -6 \end{cases}$

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} 2x + y - z = 2 \\ 3x + 3y - 2z = 3 \end{cases}$