Answer
(a) $(x,y,z) = (7, 1, 1)$
(b) $(x,y,z) = (-14,16, -2)$
Work Step by Step
(a) We can use the line of intersection to find the value of $n$:
$(x,y,z) = (0,6,0) + n(7,-5,1)$
Since $x = 7$, the $x$ coordinate in the line of intersection must be $7$:
$0+n(7) = 7$
$n = 1$
We can use the line of intersection to find the point in both planes:
$(x,y,z) = (0,6,0) + n(7,-5,1)$
$(x,y,z) = (0,6,0) + (1)(7,-5,1)$
$(x,y,z) = (0+7,6+(-5),0+1)$
$(x,y,z) = (7, 1, 1)$
(b) We can use the line of intersection to find the value of $n$:
$(x,y,z) = (0,6,0) + n(7,-5,1)$
Since $y = 16$, the $y$ coordinate in the line of intersection must be $16$:
$6+n(-5) = 16$
$n = -2$
We can use the line of intersection to find the point in both planes:
$(x,y,z) = (0,6,0) + n(7,-5,1)$
$(x,y,z) = (0,6,0) + (-2)(7,-5,1)$
$(x,y,z) = (0-14,6+10,0-2)$
$(x,y,z) = (-14,16, -2)$
