How do you solve the system #9x+3y=1, 5x+y=1# using matrix equation?

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Answer 1 · 2016-12-08T16:36:28

The answer is #x=1/3# and #y=-2/3#

Let' write the matrix corresponding to the 2 equations

#((9,3),(5,1))*((x),(y))=((1),(1))#

Let #A=((9,3),(5,1))#

Then

#((x),(y))=A^(-1)*((1),(1))#

We must calculate the inverse of matrix #A#

The inverse of matrix #((a,b),(c,d))# is

#1/(ad-bc)*((d,-b),(-c,a))#

So,

#A^(-1)=1/(-6)((1,-3),(-5,9))#

#=((-1/6,1/2),(5/6,-3/2))#

Verification

#A*A^(-1)=I#

#((-1/6,1/2),(5/6,-3/2))*((9,3),(5,1))=((1,0),(0,1))=I#

Now, we can solve our equation

#((x),(y))=((-1/6,1/2),(5/6,-3/2))*((1),(1))#

#x=-1/6+1/2=1/3#

#y=5/6-3/2=-2/3#