Question

Do the three planes, x + y - 3z = 2, 2x + y + z = 1 and 3x + 2y - 2z = 0 have a common point of intersection? If so, find one and if not, tell why there is no such point.

Answer 1

The system of equations has no solution and the three planes do not have a common point of intersection.

Step-by-step explanation:

In order for the three planes to have a common point of intersection, all three equations must be satisfied simultaneously. We can solve this system of equations by using the method of elimination.

Add the first two equations together to eliminate the y variable:
x + y - 3z + 2x + y + z = 2 + 1
Combine like terms:
3x + 2y - 2z = 3

Now we have a new system of equations:
3x + 2y - 2z = 3
3x + 2y - 2z = 0

Subtract the second equation from the first to eliminate the x variable:
(3x + 2y - 2z) - (3x + 2y - 2z) = 3 - 0
0 = 3 - 0
This is a contradiction, which means the system of equations has no solution and the three planes do not have a common point of intersection.