Get like letters, equal signs and constants lined up vertically:
(1) 2x + y = 6
(2) -x - 2y + 8z = 2
(3) -y + 4z = 3
We start by observing that z is already eliminated in (1),
and that x is already eliminated from (3).
We pick one of those, say (3).
(3) -y + 4z = 3
Since x is already eliminated from (3), we eliminate x from
the other two, (1) and (2):
(1) 2x + y = 6
(2) -x - 2y + 8z = 2
To do that we multiply (2) by 2 so that the x terms will cancel
when we add the equations term by term, getting (4):
(1) 2x + y = 6
-2x - 4y + 16z = 4
------------------------
(4) -3y + 16z = 10
Now we take (3) and (4) together as a system of 2 equations in
only 2 unknowns and line them up vertically:
(3) -y + 4z = 3
(4) -3y + 16z = 10
To do that we multiply (3) by -3 so that the y terms will cancel
when we add the equations term by term, getting (5):
3y - 12z = -9
(4) -3y + 16z = 10
-------------------
4z = 1
(5) z = 1/4
Using (5), substitute 1/4 for z in (3) to find y:
(3) -y + 4z = 3
-y + 4(1/4) = 3
-y + 1 = 3
-y = 2
(6) y = -2
Using (6), substitute -2 for y in (1)
(1) 2x + y = 6
2x + (-2) = 6
2x - 2 = 6
2x = 8
(7) x = 4
From (7), (6), and (5), we have the solution
(x,y,z) = (4,-2,1/4)
Edwin
