Questions: Use the Gauss Jordan method to solve the following system of equations. 5x - 3y + 3z = 9 5x + 2y - z = 16 15x - 4y + 5z = 34 Write the augmented matrix for the corresponding system of equations. Select the correct choice below and fill the answer boxes to complete your choice. A. C. B. [5 -3 3 9 5 2 -1 16 15 -4 5 34] D. [□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is (□, □, in the order x, y, z □ (Simplify your answers.) B. There is an infinite number of solutions. The solution is □, z ), where z is any real number. □ (Simplify your answers. Use integers or fractions for any numbers in the expressions.) C. There is no solution.

Use the Gauss Jordan method to solve the following system of equations.
5x - 3y + 3z = 9
5x + 2y - z = 16
15x - 4y + 5z = 34

Write the augmented matrix for the corresponding system of equations. Select the correct choice below and fill the answer boxes to complete your choice.
A.
C.
B.
[5 -3 3  9
5 2 -1  16
15 -4 5  34]
D. [□ □ □  □
□ □ □  □
□ □ □  □
□ □ □  □]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution is (□, □, in the order x, y, z
□
(Simplify your answers.)
B. There is an infinite number of solutions. The solution is □, z ), where z is any real number.
□
(Simplify your answers. Use integers or fractions for any numbers in the expressions.)
C. There is no solution.
Transcript text: Use the Gauss Jordan method to solve the following system of equations. \[ \begin{aligned} 5 x-3 y+3 z & =9 \\ 5 x+2 y-z & =16 \\ 15 x-4 y+5 z & =34 \end{aligned} \] Write the augmented matrix for the corresponding system of equations. Select the correct choice below and fill the answer boxes to complete your choice. A. C. B. $\left[\begin{array}{ccc|c}5 & -3 & 3 & 9 \\ 5 & 2 & -1 & 16 \\ 15 & -4 & 5 & 34\end{array}\right]$ D. $\left[\begin{array}{lll|l}\square & \square & \square \\ \square & \square & \square \\ \square & \square & \square \\ \square & \square & \square\end{array}\right]$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is ( $\square, \square$, in the order $x, y, z$ $\square$ (Simplify your answers.) B. There is an infinite number of solutions. The solution is $\square, z$ ), where $z$ is any real number. $\square$ (Simplify your answers. Use integers or fractions for any numbers in the expressions.) C. There is no solution.
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Solution

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Solution Steps

Step 1: Form the augmented matrix

The augmented matrix for the given system of equations is formed by writing the coefficients of the variables and the constants in a matrix.

\[ \begin{aligned} 5 x-3 y+3 z & =9 \\ 5 x+2 y-z & =16 \\ 15 x-4 y+5 z & =34 \end{aligned} \]

The augmented matrix is: \[ \left[\begin{array}{ccc|c} 5 & -3 & 3 & 9 \\ 5 & 2 & -1 & 16 \\ 15 & -4 & 5 & 34 \end{array}\right] \]

So, the correct choice is B.

Step 2: Perform Gauss-Jordan elimination

We perform elementary row operations to transform the augmented matrix into reduced row-echelon form.

  1. Replace R3 by R3 - 3R1: \[ \left[\begin{array}{ccc|c} 5 & -3 & 3 & 9 \\ 5 & 2 & -1 & 16 \\ 0 & 5 & -4 & 7 \end{array}\right] \]

  2. Replace R2 by R2 - R1: \[ \left[\begin{array}{ccc|c} 5 & -3 & 3 & 9 \\ 0 & 5 & -4 & 7 \\ 0 & 5 & -4 & 7 \end{array}\right] \]

  3. Replace R3 by R3 - R2: \[ \left[\begin{array}{ccc|c} 5 & -3 & 3 & 9 \\ 0 & 5 & -4 & 7 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  4. Multiply R2 by 1/5: \[ \left[\begin{array}{ccc|c} 5 & -3 & 3 & 9 \\ 0 & 1 & -4/5 & 7/5 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  5. Replace R1 by R1 + 3R2: \[ \left[\begin{array}{ccc|c} 5 & 0 & 3/5 & 66/5 \\ 0 & 1 & -4/5 & 7/5 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

  6. Multiply R1 by 1/5: \[ \left[\begin{array}{ccc|c} 1 & 0 & 3/25 & 66/25 \\ 0 & 1 & -4/5 & 7/5 \\ 0 & 0 & 0 & 0 \end{array}\right] \]

Step 3: Interpret the reduced row-echelon form

The reduced row-echelon form corresponds to the system: \[ \begin{aligned} x + \frac{3}{25}z &= \frac{66}{25} \\ y - \frac{4}{5}z &= \frac{7}{5} \end{aligned} \]

Let $z = t$, where $t$ is any real number. Then, \[ \begin{aligned} x &= \frac{66}{25} - \frac{3}{25}t \\ y &= \frac{7}{5} + \frac{4}{5}t \end{aligned} \]

Final Answer

\\(\boxed{\text{A. The solution is } (\frac{66}{25}-\frac{3}{25}z, \frac{7}{5}+\frac{4}{5}z, z), \text{where } z \text{ is any real number.} }\\)

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