Algebra 2

Solve the following system:
{2 x+4 y = 8
6 x+3 y = -3
Express the system in matrix form:
(2 | 4
6 | 3)(x
y) = (8
-3)
Write the system in augmented matrix form and use Gaussian elimination:
(2 | 4 | 8
6 | 3 | -3)
Swap row 1 with row 2:
(6 | 3 | -3
2 | 4 | 8)
Subtract 1/3 × (row 1) from row 2:
(6 | 3 | -3
0 | 3 | 9)
Divide row 1 by 3:
(2 | 1 | -1
0 | 3 | 9)
Divide row 2 by 3:
(2 | 1 | -1
0 | 1 | 3)
Subtract row 2 from row 1:
(2 | 0 | -4
0 | 1 | 3)
Divide row 1 by 2:
(1 | 0 | -2
0 | 1 | 3)
Collect results:
Answer: |  x = -2                and                y = 3

2x + 4y = 8   →   x + 2y = 4

6x + 3y  = -3  →  2x + y = -1

1  2        1   0

2  1        0   1         multiply  -2 row 1   and add to row 2

1  2        1  0

0 -3       -2  1      multiply 2/3 row 2 and add to row 1

1  0     -1/3  2/3

0 -3      -2  1         divide row 2 by -3

1  0    -1/3   2/3

0  1     2/3  -1/3

So  the inverse matrix, A^-1,  is

-1/3     2/3

 2/3   - 1/3

So  .....A^-1 b = x      and we have

[-1/3     2/3       [ 4

  2/3   -1/3 ]      [ - 1 ]     =

(-1/3)(4) + (2/3)(-1)  = -6/3  = -2   = x

(2/3) (4) + (-1/3)(-1)  = 9/3  = 3   = y

So   {x, y }  =  { -2, 3}