Solve each system of equations by using A-1. 

 x + 4y = 5
3x + 8y = 7

Write the above as the matrix equation:

[1  4][x]   [5]
[3  8][y] = [7]

Now we must find the inverse A-1 of 

    [1  4]
A = [3  8]

The inverse of a 2×2 matrix is formed by these
6 steps:

1. Find the product of the two elements in the 
   upper left to lower right diagonal. 
2. Find the product of the two elements in the 
   lower left to upper right diagonal. 
3. subtract the result of step 2 from the result
   of step 1.
4. Swap the elements in the upper left to lower 
   right diagonal.
5. Change the signs of the element in the lower 
   left to upper right diagonal.
6. Divide each element by the result of 3. This
   is A-1

Using these steps, the inverse of the 2×2 
matrix A

               [1  4]
               [3  8]

1. Find the product of the two elements in the 
   upper left to lower right diagonal.

              (1)(8) = 8

2. Find the product of the two elements in the 
   lower left to upper right diagonal. 

              (4)(3) = 12 

3. subtract the result of step 2 from the result 
   of step 1.

              8 - 12 = -4  

4. Swap the elements in the upper left to lower right diagonal.

        [1  4]         [8  4]
        [3  8] becomes [3  1]

5. Change the signs of the element in the lower left to upper right diagonal.

        [8  4]         [ 8 -4]
        [3  1] becomes [-3  1]


6. Divide each element by the result of 3.

[ 8 -4]         [ 8/-4 -4/-4]     [ -2    1]
[-3  1] becomes [-3/-4  1/-4]  or [3/4 -1/4]

That is the inverse matrix A-1.

               [ -2    1]
         A-1 = [3/4 -1/4]


Now we left-multiply both sides of the 
matrix equation

                      [1  4][x]   [5]
                      [3  8][y] = [7]

by the inverse matrix A-1

            [ -2    1][1  4][x]   [ -2    1][5]
            [3/4 -1/4][3  8][y] = [3/4 -1/4][7]

Multiply the 2×2 matrices on the left

                      [1  0][x]   [ -2    1][5]
                      [0  1][y] = [3/4 -1/4][7]

Multiply the 2×2 (identity) matrix on the left by the
2×1 matrix of variables:

                            [x]   [ -2    1][5]
                            [y] = [3/4 -1/4][7]


Multiply the 2×2 matrix by the 2×1 matrix on the right:

                            [x]   [-3]
                            [y] = [ 2]

The solution is:       x = -3 and y = 2

Edwin