Solve Using an Inverse Matrix y=x-5x+6 , y=x-2

Problem

{[y=x−5*x+6],[y=x−2])

Solution

1. Simplify the first equation by combining like terms for x

y=−4*x+6

2. Rearrange both equations into standard form a*x+b*y=c to prepare for the matrix equation.

4*x+y=6

−x+y=−2

3. Write the system as a matrix equation of the form A*X=B

[[4,1],[−1,1]]*[[x],[y]]=[[6],[−2]]

4. Calculate the determinant of the coefficient matrix A

det(A)=(4)*(1)−(1)*(−1)

det(A)=4+1=5

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

A(−1)=1/5*[[1,−1],[1,4]]

6. Solve for X by multiplying A(−1) and B

[[x],[y]]=1/5*[[1,−1],[1,4]]*[[6],[−2]]

[[x],[y]]=1/5*[[(1)*(6)+(−1)*(−2)],[(1)*(6)+(4)*(−2)]]

[[x],[y]]=1/5*[[8],[−2]]

7. Simplify the resulting values for x and y

x=8/5

y=−2/5

Final Answer

(x,y)=(8/5,−2/5)

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