# How do you solve the system x+5y=26, 3x-2y=-41 using matrix equation?

Nov 22, 2017

Write the two equations as a 2 by 3 augmented matrix.
Use elementary row operations, until an identity matrix is obtained on the left, then the column vector on the right will contain the solution.

#### Explanation:

Use the first equation, $x + 5 y = 26$, to write the first row of an augmented matrix:

[ (1,5,|,26) ]

Use the second, $3 x - 2 y = - 41$ to add the second row to the augmented matrix:

[ (1,5,|,26), (3,-2,|,-41) ]

We have the 2 by 3 augmented matrix, therefore, we may begin elementary row operations.

Multiply the first row by -3, add it to the second row, and put the result in the second row:

[ (1,5,|,26), (0,-17,|,-119) ]

Divide the second row by -17:

[ (1,5,|,26), (0,1,|,7) ]

Multiply the second row by -5 and add it to the first row:

[ (1,0,|,-9), (0,1,|,7) ]

We have an identity matrix on the left, therefore, the solution is in the column vector on the right:

$x = - 9 , y = 7$

Check:

$- 9 + 5 \left(7\right) = 26$
$3 \left(- 9\right) - 2 \left(7\right) = - 41$

$26 = 26$
$- 41 = - 41$

This checks.