Please lead me in the right direction. I am so confused and I have been working on this one problem for an hour. I have the answer, but I want to know how to get to it. The answer is (1,1,0)

Solve the system by elimination.

-2x+2y+3z=0; -2x-y+z=-3; 2x+3y+3z=5

plug in your numbers here, and see all the details:

http://www.gregthatcher.com/Mathematics/GaussJordan.aspx

To solve the system of equations by elimination, follow these steps:

1. Start by writing down the three equations:

-2x + 2y + 3z = 0 ...(Equation 1)
-2x - y + z = -3 ...(Equation 2)
2x + 3y + 3z = 5 ...(Equation 3)

2. Choose two equations and eliminate one variable at a time. Let's eliminate x first.

a. Multiply Equation 2 by -1:
2x + y - z = 3 ...(Equation 2 multiplied by -1)

b. Add Equation 1 and the modified Equation 2:
(-2x + 2y + 3z) + (2x + y - z) = 0 + 3
Simplify:
3y + 2z = 3 ...(Equation 4)

3. Now, choose a different pair of equations to eliminate another variable. Let's eliminate x again.

a. Multiply Equation 1 by 2:
-4x + 4y + 6z = 0 ...(Equation 1 multiplied by 2)

b. Add Equation 4 and the modified Equation 1:
(-4x + 4y + 6z) + (3y + 2z) = 0 + 3
Simplify:
4y + 8z = 3 ...(Equation 5)

4. We now have two equations with two variables, y and z.

5. Solve equations 4 and 5 simultaneously. Since both equations are already in terms of y and z, we can set them equal to each other:

3y + 2z = 4y + 8z

6. Rearrange the terms to get all the y variables on one side and the z variables on the other side:

3y - 4y = 8z - 2z
-y = 6z

7. Divide both sides of the equation by -1 to solve for y:

y = -6z

8. Now, substitute the value of y in terms of z into one of the previous equations. Let's use Equation 4:

3(-6z) + 2z = 3
Simplify:
-18z + 2z = 3
-16z = 3
Divide both sides by -16:
z = -3/16

9. Substitute the value of z into the equation y = -6z:

y = -6(-3/16)
Simplify:
y = 9/8

10. Now, we have the values for y and z. Substitute these values into one of the original equations. Let's use Equation 1:

-2x + 2(9/8) + 3(-3/16) = 0
Simplify:
-2x + 18/8 - 9/16 = 0
Find a common denominator:
-32x + 36 - 9/16 = 0
Combine the fractions:
-32x + 36 - 9/16 = 0
Multiply through by 16 to eliminate the fraction:
-32x(16) + 36(16) - 9 = 0
Simplify:
-512x + 576 - 9 = 0
-512x + 567 = 0
Add 512x to both sides:
567 = 512x
Divide both sides by 512:
x = 567/512

11. Therefore, the solution to the system of equations is (x, y, z) = (567/512, 9/8, -3/16). Rounding to the nearest whole numbers, the answer is approximately (1, 1, 0).