Solve Using an Inverse Matrix 2x+2y=-16 , y=-x-8

Problem

{[2*x+2*y=−16],[y=−x−8])

Solution

1. Rearrange the equations into standard form a*x+b*y=c so that the variables align.

2*x+2*y=−16

x+y=−8

2. Write the system as a matrix equation of the form A*X=B

A=[[2,2],[1,1]]

X=[[x],[y]]

B=[[−16],[−8]]

[[2,2],[1,1]]*[[x],[y]]=[[−16],[−8]]

3. Calculate the determinant of matrix A to determine if an inverse exists.

det(A)=(2)*(1)−(2)*(1)

det(A)=2−2

det(A)=0

4. Conclude that because the determinant is zero, the matrix A is singular and does not have an inverse. This indicates the system is either inconsistent or dependent.

5. Analyze the relationship between the two equations. Dividing the first equation by 2 yields x+y=−8 which is identical to the second equation. This means the lines are coincident.

Final Answer

Infinitely many solutions of the form *(x,−x−8)

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