Solve Using an Inverse Matrix 6x-12y=16 , 3x-6y=8

Problem

{[6*x−12*y=16],[3*x−6*y=8])

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=[[6,−12],[3,−6]]

X=[[x],[y]]

B=[[16],[8]]

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

det(A)=(6)*(−6)−(−12)*(3)

det(A)=−36−(−36)

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.

det(A)=0⇒A(−1)* does not exist

4. Analyze the relationship between the two equations. Divide the first equation by 2

(6*x−12*y)/2=16/2

3*x−6*y=8

5. Conclude that because the two equations are identical, the system is dependent and has infinitely many solutions.

3*x=8+6*y

x=(8+6*y)/3

Final Answer

The system has infinitely many solutions of the form *(x,y)=((8+6*y)/3,y)

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