Solve Using an Inverse Matrix 4x+3y=-2 , x-y=-4

Problem

{[4*x+3*y=−2],[x−y=−4])

Solution

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

A=[[4,3],[1,−1]]

X=[[x],[y]]

B=[[−2],[−4]]

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

det(A)=(4)*(−1)−(3)*(1)

det(A)=−4−3=−7

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

A(−1)=1/(−7)*[[−1,−3],[−1,4]]

A(−1)=[[1/7,3/7],[1/7,−4/7]]

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

X=A(−1)*B

[[x],[y]]=[[1/7,3/7],[1/7,−4/7]]*[[−2],[−4]]

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

x=(1/7)*(−2)+(3/7)*(−4)=−2/7−12/7=−14/7=−2

y=(1/7)*(−2)+(−4/7)*(−4)=−2/7+16/7=14/7=2

Final Answer

[[x],[y]]=[[−2],[2]]

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