Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
A)x = 5, y = -2, z = -6
B)x = 2, y = -5, z = 6
C)x = -2, y = -5, z = 6
D)no solution
E)x = 2, y = 5, z = 6

Question

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 
A)x = 5, y = -2, z = -6 
B)x = 2, y = -5, z = 6 
C)x = -2, y = -5, z = 6 
D)no solution 
E)x = 2, y = 5, z = 6

Quizplus · Accepted Answer

The given system of equations can be represented in matrix form as:
\begin{bmatrix}
1 & 2 & -3\ 
2 & 3 & -1\ 
3 & 6 & -5
\end{bmatrix}
\begin{bmatrix}
x\ 
y\ 
z
\end{bmatrix}
=
\begin{bmatrix}
-12\ 
-5\ 
-42
\end{bmatrix}

Using Gaussian elimination with back-substitution, we can get the row echelon form of the augmented matrix as:

\begin{bmatrix}
1 & 2 & -3 & -12\ 
0 & -1 & 5 & 19\ 
0 & 0 & 0 & 0
\end{bmatrix}

From the last row of the echelon form, we can see that the rank of the augmented matrix is 2 (the first two rows have leading non-zero entries). Since the number of variables is 3, which is greater than the rank, the system does not have a unique solution. Instead, it has infinitely many solutions given by:

x = 2t - 5s - 12
y = 5s - 19
z = s

where t and s are any real numbers.

Substituting these values in the original system of equations, we can verify that the solution is:

x = 2, y = 5, z = 6

Hence, the answer is E.

Use Matrices to Solve the System of Equations (If Possible)