Solve Using an Inverse Matrix 8x+6y=-4 , 9x-6y=-81

Problem

{[8*x+6*y=−4],[9*x−6*y=−81])

Solution

1. Write the system of equations in the matrix form A*X=B

A=[[8,6],[9,−6]]

X=[[x],[y]]

B=[[−4],[−81]]

2. Calculate the determinant of matrix A denoted as det(A)

det(A)=(8)*(−6)−(6)*(9)

det(A)=−48−54

det(A)=−102

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

A(−1)=1/(−102)*[[−6,−6],[−9,8]]

4. Solve for X by multiplying the inverse matrix A(−1) by matrix B

X=A(−1)*B

X=1/(−102)*[[−6,−6],[−9,8]]*[[−4],[−81]]

5. Perform the matrix multiplication.

X=1/(−102)*[[(−6)*(−4)+(−6)*(−81)],[(−9)*(−4)+(8)*(−81)]]

X=1/(−102)*[[24+486],[36−648]]

X=1/(−102)*[[510],[−612]]

6. Simplify the resulting vector to find the values of x and y

x=510/(−102)=−5

y=(−612)/(−102)=6

Final Answer

[[x],[y]]=[[−5],[6]]

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