Solve Using an Inverse Matrix 3x-2y=7 , 9x-6y=14

Problem

{[3*x−2*y=7],[9*x−6*y=14])

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=[[3,−2],[9,−6]]

X=[[x],[y]]

B=[[7],[14]]

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

det(A)=(3)*(−6)−(−2)*(9)

det(A)=−18−(−18)

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 consistency of the system. Since the inverse matrix method cannot be used, we check if the equations are multiples. The second equation 9*x−6*y=14 is 3(3x - 2y) = 14$.

3*(3*x−2*y)=3*(7)

9*x−6*y=21

5. Compare the equations to see if they are contradictory. The system states 9*x−6*y=14 and 9*x−6*y=21 which is impossible.

14≠21

Final Answer

No solution

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