# Thread: solving a system of linear equations

1. ## solving a system of linear equations

Solve the system of linear equations and interpret your solution geometrically:
[1] 2x + y + 2z - 4 = 0
[2] x - y - z - 2 = 0
[3] x + 2y - 6z - 12 = 0

I keep getting really odd numbers with a lot of fractions whenever i try to solve for it.
First i eliminated x
[1] 2x+y+2z-4=0 [2] x-y-z-2=0
[2] - 2x-2y-2z-4=0 [3] - x=2y-6z-12=0
[4] 3y+4z=0 [5] -3y+5z+10=0
then i eliminated y to solve for z
[4] 3y+4z=0
[5] + -3y+5z+10 = 0
9z+10=0
z=-10/9
then i solved for y
[4]3y+4(-10/9)=0
y=1.48
then i solved for x
[1]2x+1.48+2(-10/9)=0
x= 0.371

so im guessing this means my point of intersection (0.371,1.48, -1.11)
im not sure if this is correct or if i missed anything, if anyone can give me their input i would really appreciate it. I'm not sure how to interpret it geometrically either.

2. Your solution is almost correct, except in the last part, you dropped the "-4" term from equation [1] when solving for x.

The solution is $\displaystyle \left(\tfrac{64}{27},\tfrac{40}{27},-\tfrac{10}{9}\right) \approx(2.37,1.48,-1.11)$.

The important thing to take from this geometrically is that each equation is a line in 3-dimensional space. Therefore, the solution set to the system of equations represents all the points that are on all three lines simultaneously. Since you found that there is only a single solution to the system of equations, it means that all three lines intersect at a single point.