Solve Using an Inverse Matrix x-y=-2 , 7x+3y=-66

Problem

{[x−y=−2],[7*x+3*y=−66])

Solution

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

A=[[1,−1],[7,3]]

X=[[x],[y]]

B=[[−2],[−66]]

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

det(A)=(1)*(3)−(−1)*(7)

det(A)=3+7=10

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

A(−1)=1/10*[[3,1],[−7,1]]

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

X=A(−1)*B

X=1/10*[[3,1],[−7,1]]*[[−2],[−66]]

5. Perform the matrix multiplication.

X=1/10*[[(3)*(−2)+(1)*(−66)],[(−7)*(−2)+(1)*(−66)]]

X=1/10*[[−6−66],[14−66]]

X=1/10*[[−72],[−52]]

6. Simplify the resulting values for x and y

x=(−72)/10=−7.2

y=(−52)/10=−5.2

Final Answer

[[x],[y]]=[[−7.2],[−5.2]]

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