Solve Using an Inverse Matrix y=-0.5x-4 , y=x+1

Problem

{[y=−0.5*x−4],[y=x+1])

Solution

1. Rearrange the equations into standard form A*x+B*y=C so that the variables are aligned.

0.5*x+y=−4

−x+y=1

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

[[0.5,1],[−1,1]]*[[x],[y]]=[[−4],[1]]

3. Calculate the determinant of the coefficient matrix A where det(A)=a*d−b*c

det(A)=(0.5)*(1)−(1)*(−1)

det(A)=0.5+1=1.5

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

A(−1)=1/1.5*[[1,−1],[1,0.5]]

A(−1)=[[2/3,−2/3],[2/3,1/3]]

5. Solve for X by multiplying the inverse matrix by the constant matrix B such that X=A(−1)*B

[[x],[y]]=[[2/3,−2/3],[2/3,1/3]]*[[−4],[1]]

6. Compute the values for x and y through matrix multiplication.

x=(2/3)*(−4)+(−2/3)*(1)=−8/3−2/3=−10/3

y=(2/3)*(−4)+(1/3)*(1)=−8/3+1/3=−7/3

Final Answer

(x,y)=(−10/3,−7/3)

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