Question

Solve the system by elimination
{-2x+2y+3z=0
{-2x-y+Z=-3
{2x+3y+3z=5

find the solutions for x, y, z

Answer 1

To solve the system by elimination, we need to eliminate one variable at a time. Let's start by eliminating x:

1. Multiply the third equation by 2:
2(2x + 3y + 3z = 5) becomes 4x + 6y + 6z = 10.

2. Add the first equation to the modified third equation:
(-2x + 2y + 3z) + (4x + 6y + 6z) = 0 + 10.
This simplifies to 2y + 9z = 10.

Now let's eliminate x again using the second equation:

3. Multiply the second equation by -2:
-2(-2x - y + z) = -2(-3) becomes 4x + 2y - 2z = 6.

4. Add the second equation to the modified third equation:
(4x + 2y - 2z) + (2y + 9z) = 6 + 10.
This simplifies to 4x + 4y + 7z = 16.

We now have two new equations:

2y + 9z = 10
4x + 4y + 7z = 16

To eliminate y, multiply the first equation by -2:

-2(2y + 9z = 10) becomes -4y - 18z = -20.

Add the modified second equation to this equation:
(4x + 4y + 7z) + (-4y - 18z) = 16 + (-20).
This simplifies to 4x - 11z = -4.

Now we have the following equations:

2y + 9z = 10
4x - 11z = -4

To eliminate y, multiply the first equation by -2:

-2(2y + 9z = 10) becomes -4y - 18z = -20.

Add the modified second equation to this equation:
(4x + 4y + 7z) + (-4y - 18z) = 16 + (-20).
This simplifies to 4x - 11z = -4.

Now we have the following equations:

4x - 11z = -4
4x - 11z = -4

Since the equations are identical, this means there are infinitely many solutions for x, y, and z. In other words, the system is dependent and the solution set can be represented parametrically.