Final answer:
The system of equations has exactly one solution.
Step-by-step explanation:
To determine the number of solutions for the system of equations, we can use the concept of linear equations and their solutions.
Given the system of equations:
a) y + 3z = 3
b) 2 + 2y + z = 8
c) 2x + 3y - z = 1
We can rewrite the equations in a standard form:
a) x + 0y + 3z = 3
b) 2x + 2y + z = 8
c) 2x + 3y - z = 1
In order to solve this system, we need to find the coefficients of the variables x, y, and z. Then, we can determine the number of solutions using the determinant of the coefficient matrix.
The coefficient matrix is:
[1 0 3]
[2 2 1]
[2 3 -1]
The determinant of the coefficient matrix is calculated as follows:
|1 0 3|
|2 2 1|
|2 3 -1|
= (1 * (2 * (-1) - 1 * 3)) - (0 * (2 * (-1) - 2 * 3)) + (3 * (2 * 3 - 2 * 2))
= (1 * (-5)) - (0) + (3 * (6 - 4))
= -5 + 6
= 1
Since the determinant of the coefficient matrix is not equal to zero, the system of equations has a unique solution. Hence, it has exactly one solution.