Question 136248
<font size = 6 color = "red"><b>Solution by Edwin:
Here is an easier way to find the inverse of a 2x2 matrix:
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I am supposed to solve by using inverse matrices. Can you please help me? This is all one problem. On my worksheet all of the information is inside one left parenthesis vs. two as I have it shown here. 
Thank you! 

7x - 8y = -21
 x -  y = -2
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Write the system as:

{{{7x - 8y = -21}}}
{{{1x - 1y =  -2}}}

Then write that as:

{{{ 
( matrix(2,2,7,-8,1,-1) )
( matrix(2,1,x,y) ) = ( matrix(2,1,-21,-2) ) }}}

Now we must find the inverse of the 2x2 matrix {{{( matrix(2,2,7,-8,1,-1) )}}}

To find the matrix of a 2x2 determinant:

{{{( matrix(2,2,7,-8,1,-1) )}}}

1.  Find the value of its determinant:

{{{abs( matrix(2,2,7,-8,1,-1)) = (7)(-1)-(-8)(1) = -7+8 = 1}}}

2.  Swap the upper left and lower right elements of {{{( matrix(2,2,7,-8,1,-1) )}}}:

{{{( matrix(2,2,-1,-8,1,7) )}}}

3. Change the sign of the upper right and lower left elements.

{{{( matrix(2,2,-1,8,-1,7) )}}}

4. Divide every element by the value of the determinant found
   in step 1.

{{{( matrix(2,2,(-1)/1,8/1,(-1)/1,7/1) )}}}

{{{( matrix(2,2,-1,8,-1,7) )}}}

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Next we left-multiply both sides of the equation

{{{ 
( matrix(2,2,7,-8,1,-1) )
( matrix(2,1,x,y) ) = ( matrix(2,1,-21,-2) ) }}}

by the inverse {{{( matrix(2,2,-1,8,-1,7) )}}}

and we get:

{{{ 
( matrix(2,2,-1,8,-1,7) )( matrix(2,2,7,-8,1,-1) )
( matrix(2,1,x,y) ) = ( matrix(2,2,-1,8,-1,7) )( matrix(2,1,-21,-2) ) }}} 

Next we multiply the first two matrices on the left:

{{{ 
( matrix(2,2,(-1)(7)+(8
)(1),(-1)(-8)+(8)(-1),(-1)(7)+(7)(1),(-1)(-8)+(7)(-1)) )( matrix(2,1,x,y) ) = ( matrix(2,2,-1,8,-1,7) )( matrix(2,1,-21,-2) ) }}} 

Simplifying,

{{{ 
( matrix(2,2,-7+8,8-8,-7+7,8-7) )( matrix(2,1,x,y) ) = ( matrix(2,2,-1,8,-1,7) )( matrix(2,1,-21,-2) ) }}}

{{{ 
( matrix(2,2,1,0,0,1) )( matrix(2,1,x,y) ) = ( matrix(2,2,-1,8,-1,7) )( matrix(2,1,-21,-2) ) }}}

Now multiply the two matrices on the left:

{{{ 
( matrix(2,1,1x+0y,0x+1y) ) = ( matrix(2,2,-1,8,-1,7) )( matrix(2,1,-21,-2) ) }}}

Simplifying:

{{{ 
( matrix(2,1,x,y) ) = ( matrix(2,2,-1,8,-1,7) )( matrix(2,1,-21,-2) ) }}}

Now multiply the two matrices on the right:

{{{ 
( matrix(2,1,x,y) ) = ( matrix(2,1,(-1)(-21)+(8)(-2),(-1)(-21)+(7)(-2) ) )}}}

Simplifying

{{{ 
( matrix(2,1,x,y) ) = (  matrix(2,1,21-16,21-14 ))   }}}

{{{ 
( matrix(2,1,x,y) ) = (  matrix(2,1,5,7 ))   }}}

So the solution is;  {{{x=5}}},{{{y=7}}}

Edwin</pre>