Solve Using an Inverse Matrix 10x+4y=16 , 15x+6y=24

Problem

{[10*x+4*y=16],[15*x+6*y=24])

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=[[10,4],[15,6]]

X=[[x],[y]]

B=[[16],[24]]

2. Calculate the determinant of matrix A to determine if an inverse exists.

det(A)=(10)*(6)−(4)*(15)

det(A)=60−60

det(A)=0

3. Analyze the result of the determinant. Since det(A)=0 the matrix A is singular and does not have an inverse.

4. Check for consistency by simplifying the equations. Divide the first equation by 2 and the second equation by 3.

(10*x+4*y)/2=16/2⇒5*x+2*y=8

(15*x+6*y)/3=24/3⇒5*x+2*y=8

5. Identify the solution type. Because both equations simplify to the same line, the system is dependent and has infinitely many solutions.

Final Answer

The system has infinitely many solutions.

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