Solve Using an Inverse Matrix x=2y-5 , 2x-4y=-10

Problem

{[x=2*y−5],[2*x−4*y=−10])

Solution

1. Rearrange the equations into standard form a*x+b*y=c to prepare for matrix representation.

x−2*y=−5

2*x−4*y=−10

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,−2],[2,−4]]*[[x],[y]]=[[−5],[−10]]

3. Calculate the determinant of matrix A denoted as det(A) using the formula a*d−b*c

det(A)=(1)*(−4)−(−2)*(2)

det(A)=−4−(−4)

det(A)=0

4. Analyze the result of the determinant. Since det(A)=0 the matrix A is singular and does not have an inverse. This means the system is either inconsistent or dependent.

5. Check for dependency by comparing the equations. Notice that multiplying the first equation x−2*y=−5 by 2 yields the second equation 2*x−4*y=−10

6. Conclude that the system has infinitely many solutions because the two equations represent the same line.

Final Answer

The system has infinitely many solutions.

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