Question 567404
<pre>
{{{system(x+2y=4,
2x-3y=1)}}}

Write as the 2x2 coefficient matrix multiplied by the 2x1 variable 
matrix and set it equal to the 2x1 constant matrix:

{{{(matrix(2,2,1,2,2,-3))}}}{{{(matrix(2,1,x,y))}}} = {{{(matrix(2,1,4,1))}}}

To find the inverse of the 2x2 coefficient matrix: {{{(matrix(2,2,1,2,2,-3))}}}

1. Swap the upper left and lower right emements: {{{(matrix(2,2,-3,2,2,1))}}}

2. Change the signs of the upper right and lower left elements: {{{(matrix(2,2,-3,-2,-2,1))}}}

3. Calculate the determinant, either of the original matrix or the one that is the result of step 3 for their determinants are the same: 
   
original matrix's determinant:  (1)(-3)-(2)(2) = -3-4 = -7
step 3 matrix;s determinant:  (-3)(1) - (-2)(-2) = -3-4 = -7

4. Divide each term in the matrix by the value of the determinant:

{{{(matrix(2,2,3/7,2/7,2/7,-1/7))}}}

Left-multiply both sides of the matrix equation by this inverse:

{{{(matrix(2,2,3/7,2/7,2/7,-1/7))}}}{{{(matrix(2,2,1,2,2,-3))}}}{{{(matrix(2,1,x,y))}}} = {{{(matrix(2,2,3/7,2/7,2/7,-1/7))}}}{{{(matrix(2,1,4,1))}}}

{{{(matrix(2,2,expr(3/7)*1+expr(2/7)*2,expr(3/7)*2+expr(2/7)*(-3),expr(2/7)*1+expr(-1/7)*2,expr(2/7)*2+expr(-1/7)*(-3)))}}}{{{(matrix(2,1,x,y))}}} = {{{(matrix(2,1,expr(3/7)*4+expr(2/7)*1,expr(2/7)*4+expr(-1/7)*1))}}}

{{{(matrix(2,4,3/7+4/7,"","",6/7-6/7,4/7+3/7,"","",4/7+3/7))}}}{{{(matrix(2,1,x,y))}}} = {{{(matrix(2,1,12/7+2/7,8/7-1/7))}}}

{{{(matrix(2,4,7/7,"","",0/7,0/7,"","",7/7))}}}{{{(matrix(2,1,x,y))}}} = {{{(matrix(2,1,14/7,7/7))}}}

{{{(matrix(2,2,1,0,0,1))}}}{{{(matrix(2,1,x,y))}}} = {{{(matrix(2,1,2,1))}}}

{{{(matrix(2,1,1x+0y,0x+1y))}}} = {{{(matrix(2,1,2,1))}}}

{{{(matrix(2,1,x,y))}}} = {{{(matrix(2,1,2,1))}}}

x=2, y=1

Edwin</pre>