Solve Using an Inverse Matrix -(y-4)=x+9 , x-8/3y=0

Problem

{[−(y−4)=x+9],[x−8/3*y=0])

Solution

1. Rearrange the equations into standard form a*x+b*y=c

−y+4=x+9⇒−x−y=5⇒x+y=−5

x−8/3*y=0

2. Write the system as a matrix equation A*X=B

A=[[1,1],[1,−8/3]],X=[[x],[y]],B=[[−5],[0]]

3. Calculate the determinant of matrix A

det(A)=(1)*(−8/3)−(1)*(1)

det(A)=−8/3−3/3=−11/3

4. Find the inverse matrix A(−1) using the formula A(−1)=1/det(A)*[[d,−b],[−c,a]]

A(−1)=1/(−11/3)*[[−8/3,−1],[−1,1]]

A(−1)=−3/11*[[−8/3,−1],[−1,1]]=[[8/11,3/11],[3/11,−3/11]]

5. Solve for X by multiplying A(−1)*B

X=[[8/11,3/11],[3/11,−3/11]]*[[−5],[0]]

x=(8/11)*(−5)+(3/11)*(0)=−40/11

y=(3/11)*(−5)+(−3/11)*(0)=−15/11

Final Answer

[[x],[y]]=[[−40/11],[−15/11]]

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