Solve the following system of equations and give a geometrical interpretation of the result.

x + y + z = 6
2x + y − 3z = -5
4x − 5y + z = −3

x+y+z=6

2x+y-3z=-5

x-4z=-11

2x+y-3z=-5
10x+5y-15z=-25
4x-5y+z=-3
14x-14z=-28
x-z=-2
x-4z=11
3z=-13
z=-13/3
x+13/3=-2
x=-19/3
y-19/3-13/3=6
y=50/3

To solve the system of equations, we can use the method of elimination or substitution. I will explain the process of elimination.

First, let's eliminate the variable x. To do this, we can multiply the first equation by 2 and subtract it from the second equation, and multiply the first equation by 4 and subtract it from the third equation.

Equation 1: x + y + z = 6
Equation 2: 2x + y - 3z = -5
Equation 3: 4x - 5y + z = -3

Multiply Equation 1 by 2:
2(x + y + z) = 2(6)
2x + 2y + 2z = 12

Subtract Equation 2 from the result:
(2x + 2y + 2z) - (2x + y - 3z) = 12 - (-5)
2x + 2y + 2z - 2x - y + 3z = 12 + 5
y + 5z = 17 ----> Equation 4

Multiply Equation 1 by 4:
4(x + y + z) = 4(6)
4x + 4y + 4z = 24

Subtract Equation 3 from the result:
(4x + 4y + 4z) - (4x - 5y + z) = 24 - (-3)
4x + 4y + 4z - 4x + 5y - z = 24 + 3
9y + 3z = 27 ----> Equation 5

Now we have two equations:

y + 5z = 17 ----> Equation 4
9y + 3z = 27 ----> Equation 5

To eliminate z from these equations, we can multiply Equation 4 by 3 and subtract it from Equation 5:

3(y + 5z) = 3(17)
3y + 15z = 51

(9y + 3z) - (3y + 15z) = 27 - 51
9y + 3z - 3y - 15z = -24
6y - 12z = -24
Divide the equation by 6 to simplify:
y - 2z = -4 ----> Equation 6

Now we have two new equations:

y - 2z = -4 ----> Equation 6
9y + 3z = 27 ----> Equation 5

To eliminate y from these equations, we can multiply Equation 6 by 9 and subtract it from Equation 5:

9(y - 2z) = 9(-4)
9y - 18z = -36

(9y + 3z) - (9y - 18z) = 27 - (-36)
9y + 3z - 9y + 18z = 27 + 36
21z = 63
Divide the equation by 21 to find the value of z:
z = 3

Now substitute z = 3 back into Equation 6:
y - 2(3) = -4
y - 6 = -4
y = 2

Finally, substitute y = 2 and z = 3 into Equation 1 to solve for x:
x + 2 + 3 = 6
x + 5 = 6
x = 1

So the solution to the system of equations is x = 1, y = 2, and z = 3.

Now, let's consider the geometrical interpretation of the result. In three-dimensional space, the solution of these equations represents the point of intersection of three planes. Each equation can be visualized as a plane, and the solution (x, y, z) represents the point where these three planes intersect.

In this case, the three planes represented by the equations are intersecting at the point (1, 2, 3). To visualize this, imagine three flat sheets of paper representing the three planes and place them in three-dimensional space. Where these sheets intersect is the point (1, 2, 3), which is the solution to the system of equations.