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.
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.
Solution
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.
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.} }\\)