Question

Given the system of equations:

X - y - 2z = 4
-x + 3y - z = 8
-2x - y - 4z = -1
If (3, 3, -2) is a solution to the system, which point(s) is a solution to the system?
1) (4, 1, -2)
2) (2, 3, -1)
3) (3, 3, -2)
4) (1, 4, -1)

Answer 1

To determine which point(s) are solutions to the system of equations, we need to substitute each point into the three equations and check if they satisfy all three equations. The only point that is a solution to the system of equations is (3, 3, -2).

Step-by-step explanation:

To determine which point(s) are solutions to the system of equations, we need to substitute each point into the three equations and check if they satisfy all three equations. Let's start with point (3, 3, -2):

1. Equation 1: x - y - 2z = 4
2. Plugging in x = 3, y = 3, and z = -2, we have 3 - 3 - 2(-2) = 4, which is true.
3. Equation 2: -x + 3y - z = 8
4. Plugging in x = 3, y = 3, and z = -2, we have -3 + 3(3) - (-2) = 8, which is true.
5. Equation 3: -2x - y - 4z = -1
6. Plugging in x = 3, y = 3, and z = -2, we have -2(3) - 3 - 4(-2) = -1, which is true.

Therefore, the point (3, 3, -2) is a solution to the system of equations. The other points given (4, 1, -2), (2, 3, -1), and (1, 4, -1) can also be substituted into the equations to check if they satisfy all three equations. We find that none of these points satisfy all three equations. Hence, the only point that is a solution to the system of equations is (3, 3, -2).