Final answer:
The system of equations is inconsistent, with two equations that can be solved independently of the third, which does not contain all variables. This leads to infinitely many solutions for z, with x and y being determined by the first two equations. The system seems underdetermined for the given variables.
Step-by-step explanation:
To identify the number of solutions to the given system of linear equations, we need to make sure those equations are written in standard form. Reformulating the equations for clarity, we get:
- 2 + 2y + 32 = 4 -> 2y + 34 = 4 -> 2y = -30 -> y = -15
- -3x + 2y - 2 = 12 -> -3x + 2(-15) - 2 = 12 -> -3x - 32 = 12 -> -3x = 44 -> x = -44/3
- -2x - 2y - 4z = -14
However, when we try to solve these equations as a system, we see that the first two equations do not contain all three variables and can be solved independently of the third, which is inconsistent. Therefore, the system of equations, as given, does not provide enough information to determine a unique solution for x, y, and z.
Based on the information provided, two of the equations can be solved directly without reference to z, but without additional constraints or equations involving z, we cannot determine a unique solution for the third variable. Consequently, the system as described does not fit with canonical systems of linear equations, which typically have an equal number of equations and unknowns. Here, with two independent equations for two variables and a third incompletely specified equation, the system is underdetermined. Thus, the answer in this case would be Option C: Infinitely many solutions for the variable z, assuming x and y are fixed by the first two equations. Without the correct context, this could also imply an error in the provided equations.