Solve Using an Inverse Matrix -5x+y=-8 , y=-4x+1

Problem

{[−5*x+y=−8],[y=−4*x+1])

Solution

1. Rearrange the equations into standard form a*x+b*y=c so that the variables are aligned.

−5*x+y=−8

4*x+y=1

2. Write the system as a matrix equation A*X=B where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

[[−5,1],[4,1]]*[[x],[y]]=[[−8],[1]]

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

det(A)=(−5)*(1)−(1)*(4)

det(A)=−5−4=−9

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

A(−1)=1/(−9)*[[1,−1],[−4,−5]]

A(−1)=[[−1/9,1/9],[4/9,5/9]]

5. Solve for X by multiplying both sides by the inverse matrix: X=A(−1)*B

[[x],[y]]=[[−1/9,1/9],[4/9,5/9]]*[[−8],[1]]

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

x=(−1/9)*(−8)+(1/9)*(1)=8/9+1/9=1

y=(4/9)*(−8)+(5/9)*(1)=−32/9+5/9=−27/9=−3

Final Answer

[[x],[y]]=[[1],[−3]]

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