Final answer:
The system of equations is unsolvable as-is because the first two equations are not independent, making it impossible to definitively solve for three variables with only two unique equations. The problem must be corrected to include three distinct equations to find a solution for the variables x, y, and z.
Step-by-step explanation:
To solve the system using a matrix, we must first note that there is an issue with the given equations. The first two equations are essentially the same, which makes it impossible to solve for three variables with only two distinct equations. Therefore, it is impossible to find a definitive solution for the system 3x + 2y = 13, 3x + 2y + 2 = 13, and 2y + 3z = 9 as it stands. In a solvable system, there should be three different equations to solve for x, y, and z.
However, if we ignore the second equation, we are left with two equations. The second equation can be rewritten without the constant 2, and then the system would not have a unique solution. Looking at the remaining two equations, we could solve for y and z, but not x without another independent equation involving x.
In a situation where we have the appropriate number of independent equations, we would convert the equations into an augmented matrix, perform row operations to reach row-echelon form or reduced row-echelon form, and then back substitute to find the values of x, y, and z.