Solve Using an Inverse Matrix 2x+y=4 , -6x-3y=-12

Problem

{[2*x+y=4],[−6*x−3*y=−12])

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=[[2,1],[−6,−3]]

X=[[x],[y]]

B=[[4],[−12]]

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

det(A)=(2)*(−3)−(1)*(−6)

det(A)=−6−(−6)

det(A)=0

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

4. Analyze the equations to determine the nature of the solutions. Divide the second equation −6*x−3*y=−12 by −3

(−6*x−3*y)/(−3)=(−12)/(−3)

2*x+y=4

5. Conclude that because both equations are identical, the system is dependent and has infinitely many solutions. We can express y in terms of x

y=4−2*x

Final Answer

Infinitely many solutions of the form *(x,4−2*x)

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