Given the equations:
color(white)("XXX")x-6y=31XXXx−6y=31
color(white)("XXX")6x+9y=-84XXX6x+9y=−84
We can write these in "augmented matrix form as:)
color(white)("XXX")(
(1,-6,31),(6,9,-84)
)
This can be solved using the normal operations we would perform on the original equations (with the variables and equal side "assumed").
color(red)("Alternately, we can use Cramer's Rule with Matrix Determinants")
color(white)("XXX")color(blue)("Although I demonstrate the soloution (below) using hand calculations")
color(white)("XXX")color(blue)("the power of the Matrix Methods lies in their compatibility with")
color(white)("XXX")color(blue)("computer systems. For example, the evaluation of
Determinants")
color(white)("XXX")color(blue)("is a build-in function for most spreadsheets.")
If M_(xy)=((1,-6),(6,9))color(white)("XX")M_(cy)=((31,-6),(-84,9))color(white)("XX")M_(xc)=((1,31),(6,-84))
x=(|M_(cy)|)/(|M_(xy)|)" and " y=(|M_xc|)/(|M_(xy)|)
Where the Determinants:
color(white)("XXX")|M_(xy)|= 1xx9-6xx(-6)= 9+36=45
color(white)("XXX")|M_(cy)|= 31xx9-(-84)xx(-6))=279-504=-255
color(white)("XXX")|M_(xc)|=1xx(-84)-6xx31=-84-186=-270
Giving
color(white)("XXX")x=-255/45=-6
and
color(white)("XXX")y=-270/45=-5
