SOLUTION: x+y+z=6 2x-y+z=8 x-2y-z=-19 I need help solving these systems of equations.

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: x+y+z=6 2x-y+z=8 x-2y-z=-19 I need help solving these systems of equations.      Log On


   

Question 471629: x+y+z=6
2x-y+z=8
x-2y-z=-19
I need help solving these systems of equations.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

x%2By%2Bz=6.......1
2x-y%2Bz=8...........2
x-2y-z=-19......3
--------------------------------
x%2By%2Bz=6.......1
x-2y-z=-19......3......subtract 3 from 1 to eliminate x
------------------------
x-x%2By-%28-2y%29%2Bz-%28-z%29=6-%28-19%29

y%2B2y%2Bz%2Bz=6%2B19
3y%2B2z=25.............a

x%2By%2Bz=6.......1
2x-y%2Bz=8...........2
--------------------------......subtract 2 from 1 to eliminate z
x-2x%2By-%28-y%29%2Bz-z=6-8

-x%2By%2By=-2
-x%2B2y=-2.............b

3y%2B2z=25.............a.....solve for z

2z=25-3y
z=25%2F2-3y%2F2

x%2By%2Bz=6.......1..plug in z
x%2By%2B25%2F2-3y%2F2=6.........c

now take -x%2B2y=-2.....b ...solve for x
2y%2B2=x.......plug it in c

x%2By%2B25%2F2-3y%2F2=6.........c
2y%2B2%2By%2B25%2F2-3y%2F2=6....solve for y
3y%2B2%2B25%2F2-3y%2F2=6....both sides multiply by 2
3y%2A2%2B2%2A2%2B%2825%2F2%292-%283y%2F2%292=6%2A2
6y%2B4%2B25-3y=12
3y%2B29=12
3y=12-29
3y=-17
y=-17%2F3
y=-5.7
now plug it in .....b and find x
-x%2B2%28-5.7%29=-2
-x-11.4=-2

-x=-2%2B11.4
-x=9.4
x=-9.4........now find z


z=25%2F2-3y%2F2
z=25%2F2-3%28-5.7%29%2F2
z=12.5%2B8.55
z=21.05
so, solutions are:

x=-9.4
y=-5.7
z=21.05
check:
x%2By%2Bz=6.......1
-9.4-5.7%2B21.05=6
-15.1%2B21.05=6
5.95=6.....round it
6=6