Solve Using an Inverse Matrix -2x=27-5y , 4y=20+2x

Problem

{[−2*x=27−5*y],[4*y=20+2*x])

Solution

1. Rearrange the equations into standard form a*x+b*y=c by moving all variable terms to the left side and constants to the right side.

−2*x+5*y=27

−2*x+4*y=20

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.

[[−2,5],[−2,4]]*[[x],[y]]=[[27],[20]]

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

det(A)=(−2)*(4)−(5)*(−2)

det(A)=−8+10=2

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

A(−1)=1/2*[[4,−5],[2,−2]]

A(−1)=[[2,−2.5],[1,−1]]

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

[[x],[y]]=[[2,−2.5],[1,−1]]*[[27],[20]]

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

x=(2)*(27)+(−2.5)*(20)=54−50=4

y=(1)*(27)+(−1)*(20)=27−20=7

Final Answer

[[x],[y]]=[[4],[7]]

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