Solve Using an Inverse Matrix x+y=19 , x-4y=-1

Problem

{[x+y=19],[x−4*y=−1])

Solution

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

A=[[1,1],[1,−4]]

X=[[x],[y]]

B=[[19],[−1]]

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

det(A)=(1)*(−4)−(1)*(1)

det(A)=−4−1=−5

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

A(−1)=1/(−5)*[[−4,−1],[−1,1]]

A(−1)=[[0.8,0.2],[0.2,−0.2]]

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

X=[[0.8,0.2],[0.2,−0.2]]*[[19],[−1]]

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

x=(0.8)*(19)+(0.2)*(−1)=15.2−0.2=15

y=(0.2)*(19)+(−0.2)*(−1)=3.8+0.2=4

Final Answer

[[x],[y]]=[[15],[4]]

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