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

Problem

{[5*y=x−1],[x−y=1])

Solution

1. Rearrange the equations into standard form a*x+b*y=c to align the variables.

x−5*y=1

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.

[[1,−5],[1,−1]]*[[x],[y]]=[[1],[1]]

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

det(A)=(1)*(−1)−(−5)*(1)

det(A)=−1+5=4

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

A(−1)=1/4*[[−1,5],[−1,1]]

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

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

6. Perform the matrix multiplication.

[[x],[y]]=1/4*[[(−1)*(1)+(5)*(1)],[(−1)*(1)+(1)*(1)]]

[[x],[y]]=1/4*[[4],[0]]

7. Simplify the resulting vector to find the values of x and y

[[x],[y]]=[[1],[0]]

Final Answer

(x,y)=(1,0)

Want more problems? Check here!