(2,-6,1)
There are a few way to solve this with linear programming, but I am using a simple substitution method.
The goal is to isolate the variables by subtracting or adding the equations. (note: I will refer to the equations as A,B, and C)
2x+y−z=−3
5x−2y+2z=24
3x−z=5
C already has just two variables, x and z. This means a good place to start is by eliminating at least y from one other equation, if not y and another value. To do this, we need to either add two equations where the y values are opposite or subtract one where the y value is equal. However, the two equations with a y value do not have opposite or the same y values.
To get a new equation, we can multiply A by 2. This will give us +2y, which can be added to B to eliminate the value all together- AND the z value. Remember that the WHOLE A equation needs to be multiplied by 2:
2(2x+y-z)=2(-3)
4x+2y-2z=-6
We can now add 2A to B.
(4x+2y-2z=-6)+(5x−2y+2z=24
)
9x=18
x=2
We now know x=2. We can plug this into C to find the z value.
3x−z=5
3(2)-z=5
6-z=5
-z=-1
z=1
With both x and z, we can find y using A.
2x+y−z=−3
2(2)+y-(1)=-3
4+y-1=-3
3+y=-3
y=-6
x=2, y=-6, z=1
