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