Solve Using an Inverse Matrix -5x+y=7 , x+y=1

Problem

{[−5*x+y=7],[x+y=1])

Solution

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

A=[[−5,1],[1,1]]

X=[[x],[y]]

B=[[7],[1]]

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

det(A)=(−5)*(1)−(1)*(1)

det(A)=−5−1=−6

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

A(−1)=1/(−6)*[[1,−1],[−1,−5]]

A(−1)=[[−1/6,1/6],[1/6,5/6]]

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

X=[[−1/6,1/6],[1/6,5/6]]*[[7],[1]]

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

x=(−1/6)*(7)+(1/6)*(1)=−7/6+1/6=−6/6=−1

y=(1/6)*(7)+(5/6)*(1)=7/6+5/6=12/6=2

Final Answer

[[x],[y]]=[[−1],[2]]

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