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

Problem

{[10*x+5*y=4],[−2*x=y+4])

Solution

1. Rearrange the second equation into standard form a*x+b*y=c by adding y to both sides.

−2*x−y=4

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.

[[10,5],[−2,−1]]*[[x],[y]]=[[4],[4]]

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

det(A)=(10)*(−1)−(5)*(−2)

det(A)=−10−(−10)

det(A)=0

4. Determine the solvability of the system. Since the determinant is 0 the matrix A is singular and does not have an inverse.

det(A)=0⇒A(−1)* does not exist

5. Analyze the equations to see if the system is inconsistent or dependent. Multiply the second equation −2*x−y=4 by −5

−5*(−2*x−y)=−5*(4)

10*x+5*y=−20

6. Compare the resulting equation to the first equation 10*x+5*y=4 Since 4≠−20 the lines are parallel and never intersect.

10*x+5*y=4

10*x+5*y=−20

Final Answer

No solution

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