Explain how to solve the following system of equations. What is the solution to the new system?

2x + 2y + z = -5

3x + 4y + 2z = 0

x + 3y + 2c = 1

I will assume you have a typo in the last equation, and that 2c is 2z

double the first:
4x + 4y + 2z = -10 ----> *
subtract the end:

x = -10 , well , that was lucky

4x + 4y + 2z = -10
x + 3y + 2c = 1
subtract them:
3x + y = -11
sub in x = -10
-30 + y = -11
y = 19

back in the original first:
-20 + 38 + z = -5
z = -23

I tested the values, they work in all 3 equations

I did make a typo lol

And thanks a ton, very nicely structured too! Thank you for the help.

To solve the system of equations, we can use the method of Gaussian elimination. This involves using various operations to manipulate the equations and simplify them into a form where the values of the variables can be easily determined.

Step 1: Write the system of equations in matrix form
We can rewrite the system of equations as a matrix equation:

[2 2 1] [x] [-5]
[3 4 2] * [y] = [ 0]
[1 3 2] [z] [ 1]

Step 2: Apply row operations to simplify the system
We can perform row operations to simplify the system of equations. Our goal is to create zeros below the main diagonal of the matrix.

First, let's subtract 1.5 times the first row from the second row to eliminate the x term in the second equation:
[2 2 1] [x] [-5]
[0 1 -0.5] * [y] = [7.5]
[1 3 2] [z] [ 1]

Next, subtract 0.5 times the first row from the third row to eliminate the x term in the third equation:
[2 2 1] [x] [-5]
[0 1 -0.5] * [y] = [7.5]
[0 2 1] [z] [ 6]

Next, multiply the second row by -2 and add it to the third row to eliminate the y term in the third equation:
[2 2 1] [x] [-5]
[0 1 -0.5] * [y] = [7.5]
[0 0 2] [z] [-6]

Step 3: Solve for the variables
The system is now in its simplified form:

2x + 2y + 1z = -5
y - 0.5z = 7.5
2z = -6

From the third equation, we can immediately determine that z = -3. Then, substituting this value into the second equation, we have:

y - 0.5(-3) = 7.5
y + 1.5 = 7.5
y = 6

Finally, substituting the values of y and z into the first equation, we have:

2x + 2(6) + (-3) = -5
2x + 12 - 3 = -5
2x + 9 = -5
2x = -14
x = -7

So, the solution to the system of equations is x = -7, y = 6, and z = -3.