Solve an x+y+z equation?

Question

X + 2Y - Z = 5
2X - 4Y + Z = 0
3X + 2Y + 2Z = 3

OK I know there is a long way and a short way. Please show me the short way. Thankyou.

Answer 1

z=4y-2x
x+2y-(4y-2x)=5
x+2x+2y-4y=5
3x-2y=5

and

3x+2y+2(4y-2x)=3
3x-4x+2y+8y=3
-x+10y=3

5(3x-2y=5)
15x-10y=25
+ -x+10y=3
14x=28
x=2

3(2)-2y=5
6-2y=5
2y=-1
y=-1/2

z=4(-1/2)-2(2)
z=-2-4
z=-6

so, x=2, y=-1/2, and z=-6

sorry but this is the only way i know how to do this. hope it helps =).

Answer 2

The short way is to have a calculator or spreadsheet that can handle the work when you enter the coefficients. Doing it that way, x = 2, y = 1/2, z = -2.

The slightly longer way is to add the first two equations to get: 3x - 2y = 5. Then twice the 1st to the 3rd to get: 5x + 6y = 13. Then add thrice the 3x - 2y = 5 to the 5x + 6y = 13 to get: 14x = 28 and then: x = 2. Substitute back into either: 3*2 - 2y = 5 so: 6 - 2y = 5 so: -2y = -1 so: y = 1/2. Now pick any of the original equations and substitute: 2 + 2*(1/2) -z = 5 so: 2 + 1 - z = 5 so: -z = 2 and finally z = -2. Bing-bing, bang, bang, boom!

Answer 3

Add equations 1 and 2:-
3x - 2y = 5

2x + 4y - 2z = 10
3x + 2y + 2z = 3----ADD

5x + 6y = 13
9x - 6y = 15----ADD
14x = 28
x = 2

10 + 6y = 13
6y = 3
y = 1/2

2x - 4y + z = 0
4 - 2 + z = 0
z = - 2

SOLUTION
x = 2 , y = 1 / 2 , z = - 2

Answer 4

eqn 1 + eqn 2 gives you
X + 2Y -Z + 2X -4Y +Z = 5 + 0
3x - 2y = 5
x=(5+2y)/3
2x eqn1 + eqn3 gives you
5x + 6y = 13 where you can plug in your x=(5+2y)/3 into and get a value for y, which is 1/2
x is therefore 2 (from plugging y=1/2 into any other eqn with just x and y)
and then you plug those values into one of your original equations to find z, which is -2