Solve Using an Inverse Matrix 2x+5y=1 , 6x+15y=3

Problem

{[2*x+5*y=1],[6*x+15*y=3])

Solution

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

A=[[2,5],[6,15]]

X=[[x],[y]]

B=[[1],[3]]

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

det(A)=(2)*(15)−(5)*(6)

det(A)=30−30

det(A)=0

3. Determine the existence of the inverse matrix A(−1) Since the determinant is 0 the matrix A is singular and does not have an inverse.

4. Analyze the relationship between the two equations. Notice that the second equation 6*x+15*y=3 is exactly 3 times the first equation 2*x+5*y=1

3*(2*x+5*y)=3*(1)

6*x+15*y=3

5. Conclude the nature of the solution. Because the equations are dependent (they represent the same line), there are infinitely many solutions that satisfy the relation 2*x+5*y=1

Final Answer

The system has infinitely many solutions.

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