Question

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

Answer 1

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+5y=26`, to write the first row of an augmented matrix:

`[ (1,5,|,26) ]`

Use the second, `3x-2y=-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(7)=26`
`3(-9)-2(7)=-41`

`26 = 26`
`-41 = -41`

This checks.