2x - 2y + z= -7
-x + 3y + 2z = 0
x - 4y - 3z= 1

Step-by-step explanation:

I used Cramer's Rule to find y, since we can isolate and solve for individual variables using Cramer's Rule.  Do that by finding the determinant of the matrix, |A|.  The matrix A looks like this:

2   -2   1

-1     3   2

1    -4   -3

and we find the determinant of a 3x3 by expanding it.  Do that by picking up the first 2 columns and throw them on at the end, like this:

2   -2    1     2   -2

-1     3    2    -1    3

1    -4    -3    1    -4

and find the determinant by multiplying along the 3 major axes and subtract from that the product of the 3 minor axes:

[(2*3*-3)+(-2*2*1)+(1*-1*-4)] - [(1*3*1)+(-4*2*2)+(-3*-1*-2)] which simplifies to

-18 - (-19) = 1

So the determinant of the matrix A is |A| = 1.

Now to find the determinant of Ay, we replace the y column with the solutions and so the same thing by expansion and then multiplying and subtracting:

2    -7     1     2     -7

-1      0     2    -1     0

1      1     -3     1      1

and find the determinant of y:

[(2*0*-3)+(-7*2*1)+(1*-1*1)] - [(1*0*1)+(1*2*2)+(-3*-1*-7)] which simplifies to

-15 - (-17) = 2

So the detminant of y is |Ay| = 2

We can solve for the variable now by dividing Ay by A:

2 / 1 = 2

So the solution for y = 2