Solve Using an Inverse Matrix -3x-4y=2 , 8y=-6x-4

Problem

{[−3*x−4*y=2],[8*y=−6*x−4])

Solution

1. Rearrange the second equation into standard form A*x+B*y=C by adding 6*x to both sides.

6*x+8*y=−4

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

[[−3,−4],[6,8]]*[[x],[y]]=[[2],[−4]]

3. Calculate the determinant of the coefficient matrix A where A=[[a,b],[c,d]]

det(A)=a*d−b*c

det(A)=(−3)*(8)−(−4)*(6)

det(A)=−24−(−24)=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 check for consistency. Divide the second equation 6*x+8*y=−4 by −2

−3*x−4*y=2

The two equations are identical, meaning the system is dependent and has infinitely many solutions.

Final Answer

The system has infinitely many solutions because *det(A)=0

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