3x+2y+4z=12
x+y+2z=6
(x=0,y=4,z=1)
determine if the given 2 lines intersect at the given point. explain your reasoning.
-2x-4y+z=8
4x+2y=-5
(x=-2,y=0,z=3) same as up above

Substitute the given points into each equation and if the answer is as intended for both equation, then it means the given points are the intersection point

Given points x = 0, y = 4, and z = 1
Equation 1:
3x + 2y + 4z = 3(0) + 2(4) + 4(1) = 0 + 8 + 4 = 12
Equation 2: 
x + y + 2z = 0 + 4 + 2(1) = 6

Both equation 1 and equation 2 give the intended answer; 12 and 6 respectively, hence the given point is the intersection point


Given point : x = -2, y = 0, and z = 3

Equation 1:
-2x - 4y + z = -2(-2) - 4(0) + 3 = 4 - 0 + 3 = 7
Equation 2:
4x + 2y = 4(-2) + 2(0) = -8

The answers are not as intended, hence the given point is not the intersection point