Solve Using an Inverse Matrix 2x+4y=25 , x+5y=2

Problem

{[2*x+4*y=25],[x+5*y=2])

Solution

1. Write the system in matrix form A*X=B where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

A=[[2,4],[1,5]]

X=[[x],[y]]

B=[[25],[2]]

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

det(A)=(2)*(5)−(4)*(1)

det(A)=10−4=6

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

A(−1)=1/6*[[5,−4],[−1,2]]

A(−1)=[[5/6,−2/3],[−1/6,1/3]]

4. Solve for X by multiplying the inverse matrix by the constant matrix, X=A(−1)*B

X=[[5/6,−2/3],[−1/6,1/3]]*[[25],[2]]

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

x=(5/6)*(25)+(−2/3)*(2)=125/6−4/3=125/6−8/6=117/6=39/2

y=(−1/6)*(25)+(1/3)*(2)=−25/6+2/3=−25/6+4/6=−21/6=−7/2

Final Answer

[[x],[y]]=[[39/2],[−7/2]]

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