Solve Using an Inverse Matrix 8x-5y=3 , 16x-10y=6

Problem

{[8*x−5*y=3],[16*x−10*y=6])

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=[[8,−5],[16,−10]]

X=[[x],[y]]

B=[[3],[6]]

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

det(A)=(8)*(−10)−(−5)*(16)

det(A)=−80−(−80)

det(A)=0

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

A(−1)* does not exist

4. Analyze the relationship between the equations. Observe that the second equation 16*x−10*y=6 is exactly 2 times the first equation 8*x−5*y=3

2*(8*x−5*y)=2*(3)

16*x−10*y=6

5. Conclude the nature of the solution. Because the equations are dependent and represent the same line, there are infinitely many solutions. We can express x in terms of y

8*x=5*y+3

x=(5*y+3)/8

Final Answer

Infinitely many solutions of the form *(x,y)=((5*y+3)/8,y)

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