Question 176914
<font size = 7 color = "red"><b>Edwin's solution:</pre></b></font>
<pre><font size = 4 color = "indigo"><b>
{{{system(5x-12y = 4,4x-7y=-2)}}}

Write this:

{{{matrix(1,3,x,"=",abs(matrix(2,2,__,__,__,__))/abs(matrix(2,2,__,__,__,__)))}}} and {{{matrix(1,3,y,"=",abs(matrix(2,2,__,__,__,__))/abs(matrix(2,2,__,__,__,__)))}}}

First fill in both denominators with the
coefficients as they appear left of the equal signs, like this{{{matrix(2,2,5,-12,4,-7)}}}

{{{matrix(1,3,x,"=",abs(matrix(2,2,__,__,__,__))/abs(matrix(2,2,5,-12,4,-7)))}}} and {{{matrix(1,3,y,"=",abs(matrix(2,2,__,__,__,__))/abs(matrix(2,2,5,-12,4,-7)))}}}

Now since {{{x}}} appears FIRST in the problem, fill in
the FIRST column for x with the numbers on the right of the
equal sign, namely these: {{{matrix(2,1,4,-2)}}}

{{{matrix(1,3,x,"=",abs(matrix(2,2,4,__,-2,__))/abs(matrix(2,2,5,-12,4,-7)))}}} and {{{matrix(1,3,y,"=",abs(matrix(2,2,__,__,__,__))/abs(matrix(2,2,5,-12,4,-7)))}}}

Now since {{{y}}} appears SECOND in the problem, fill in
the SECOND column for y with the numbers on the right of the
equal sign, namely these: {{{matrix(2,1,4,-2)}}}

{{{matrix(1,3,x,"=",abs(matrix(2,2,4,__,-2,__))/abs(matrix(2,2,5,-12,4,-7)))}}} and {{{matrix(1,3,y,"=",abs(matrix(2,2,__,4,__,-2))/abs(matrix(2,2,5,-12,4,-7)))}}}

Complete the determinants by bringing the columns below up
to the top:

{{{matrix(1,3,x,"=",abs(matrix(2,2,4,-12,-2,-7))/abs(matrix(2,2,5,-12,4,-7)))}}} and {{{matrix(1,3,y,"=",abs(matrix(2,2,5,4,4,-2))/abs(matrix(2,2,5,-12,4,-7)))}}}

To evaluate {{{abs(matrix(2,2,A,B,C,D))}}}, use this formula.
which amounts to subtracting the diagonal products:

 {{{abs(matrix(2,2,A,B,C,D)) = AD - BC}}} 

{{{matrix(1,5,x,"=",abs(matrix(2,2,4,-12,-2,-7))/abs(matrix(2,2,5,-12,4,-7)),"=",((4)(-7)-(-12)(-2))/((5)(-7)-(-12)(4)))}}}

{{{matrix(1,13,x,"=",abs(matrix(2,2,4,-12,-2,-7))/abs(matrix(2,2,5,-12,4,-7)),"=",((4)(-7)-(-12)(-2))/((5)(-7)-(-12)(4)),"=",((-28)-(24))/((-35)-(-48)),"=",(-52)/(-35+48),"=",(-52)/13, "=",-4)}}}

Now for y.  We only have to do the top
determinant because we have already done
the bottom determinant, and found it to be 13, so

{{{matrix(1,11,y,"=",abs(matrix(2,2,5,4,4,-2))/abs(matrix(2,2,5,-12,4,-7)),"=",((5)(-2)-(4)(4))/13, "=", ((-10)-(16))/13, "=",(-26)/13, "=", -2)}}}

Edwin</pre>