2x+2y+3z=-1
4x+4y+3z=-5
4x+2y+2z=0
To solve this system of equations, we can use the method of elimination.
First, we will eliminate the x term by multiplying the first equation by -2 and the third equation by 2:
-4x - 4y - 6z = 2
4x + 2y + 2z = 0
Adding these two equations together, we get:
-2y - 4z = 2 (Equation 1)
Now, we will eliminate the x term by multiplying the second equation by -2:
-8x - 8y - 6z = 10
Adding this new equation to the first equation we obtained, Equation 1, we get:
-10y - 10z = 12 (Equation 2)
Next, multiply the second equation by -2 to eliminate the x term:
-8x - 4y - 4z = 0 (Equation 3)
Adding Equation 3 to the third equation, we get:
-2y - 2z = 0 (Equation 4)
Now, we have a system of two equations with two variables: Equation 2 and Equation 4. We can solve this system to find the values of y and z.
Multiplying Equation 4 by -5:
10y + 10z = 0 (Equation 5)
Adding Equation 5 to Equation 2:
0 + 20z = 12
20z = 12
z = 12/20
z = 3/5
Substituting the value of z back into Equation 4:
-2y - 2(3/5) = 0
-2y - 6/5 = 0
-2y = 6/5
y = -3/5
Now, we can substitute the values of y and z back into one of the original equations to solve for x. Let's use the third equation:
4x + 2(-3/5) + 2(3/5) = 0
4x - 6/5 + 6/5 = 0
4x = 0
x = 0
Therefore, the solution to the system of equations is x = 0, y = -3/5, and z = 3/5.