Solve Using an Inverse Matrix -4x+9y=41 , 5x-4y=-44

Problem

{[−4*x+9*y=41],[5*x−4*y=−44])

Solution

1. Write the system in matrix form A*X=B where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

A=[[−4,9],[5,−4]]

X=[[x],[y]]

B=[[41],[−44]]

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

|A|=(−4)*(−4)−(9)*(5)

|A|=16−45

|A|=−29

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

A(−1)=1/(−29)*[[−4,−9],[−5,−4]]

A(−1)=[[4/29,9/29],[5/29,4/29]]

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

X=[[4/29,9/29],[5/29,4/29]]*[[41],[−44]]

5. Perform the matrix multiplication to find the values of x and y

x=4/29*(41)+9/29*(−44)=(164−396)/29=(−232)/29=−8

y=5/29*(41)+4/29*(−44)=(205−176)/29=29/29=1

Final Answer

[[x],[y]]=[[−8],[1]]

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