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

1 Answer
Nov 22, 2017

Answer:

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.