Solve the system by elimination.
-2x+2y+3z=0
-2x-y+z=-3
2x+3y+3z=5
visit
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
for all the details.
To solve the system of equations by elimination, we want to eliminate one variable at a time by adding or subtracting the equations.
Let's start by eliminating the variable "x."
The first equation is -2x + 2y + 3z = 0.
The second equation is -2x - y + z = -3.
To eliminate x, we can add the two equations together. The -2x terms will cancel each other out.
(-2x + 2y + 3z) + (-2x - y + z) = (0) + (-3).
Simplifying the left side:
-2x + -2x + 2y - y + 3z + z = -3.
-4x + y + 4z = -3.
Now, let's move on to eliminating the variable "y."
The first equation is -2x + 2y + 3z = 0.
The third equation is 2x + 3y + 3z = 5.
To eliminate y, we can multiply both sides of the first equation by 3 and the second equation by 2, and then subtract the equations.
(3) * (-2x + 2y + 3z) + (2) * (2x + 3y + 3z) = (3) * 0 + (2) * 5.
Simplifying the left side:
-6x + 6y + 9z + 4x + 6y + 6z = 0 + 10.
-2x + 12y + 15z = 10.
Now, we have two equations:
-4x + y + 4z = -3,
-2x + 12y + 15z = 10.
To eliminate the variable "x," we can multiply both sides of the first equation by -2 and the second equation by -4, and then add the equations together.
(-2) * (-4x + y + 4z) + (-4) * (-2x + 12y + 15z) = (-2) * (-3) + (-4) * 10.
Simplifying the left side:
8x - 2y - 8z + 8x - 48y - 60z = 6 - 40.
16x - 50y - 68z = -34.
Now, we are left with one equation:
16x - 50y - 68z = -34.
This is the final equation obtained through the elimination method.