SOLUTION: PLEASE HELP! For each system of equations, use determinants (D and possibly D small y) to state how many solutions exist. Then circle the appropriate conclusions about the equation

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Question 1023408: PLEASE HELP! For each system of equations, use determinants (D and possibly D small y) to state how many solutions exist. Then circle the appropriate conclusions about the equations and graphs.
3x+2y=5
-x+5y=7
-x-2y=-3
3x+6y=9
Equations are: consistent, inconsistent, or dependent
Graph of the lines: intersect, are parallel, or coincide
Answer by ikleyn(53937) About Me  (Show Source):
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PLEASE HELP! For each system of equations, use determinants (D and possibly D small y) to state how many solutions exist.
Then circle the appropriate conclusions about the equations and graphs.
3x+2y=5
-x+5y=7
-x-2y=-3
3x+6y=9
Equations are: consistent, inconsistent, or dependent
Graph of the lines: intersect, are parallel, or coincide
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1. It would be good if you subdivide the set of your equations in two separate systems before posting.
   Otherwise I can think I don't know what.


2. 
system%283x%2B2y=5%2C%0D%0A-x%2B5y=7%29.

Determinant D = det %28matrix%282%2C2%2C+3%2C2%2C+-1%2C5%29%29 = 3*5 - 2*(-1) = 15 + 2 = 17.

Since the determinant is not zero, the system is consistent.
The lines intersect.


3.
system%28-x-2y=-3%2C%0D%0A3x%2B6y=9%29.

Determinant D = det %28matrix%282%2C2%2C+-1%2C-2%2C+3%2C6%29%29 = (-1)*6 - (-2)*3 = -6 + 6 = 0.

This is a signal that the matrix rows are proportional.
They really are proportional.
Notice that the right sides are proportional with the same proportionality coefficient.

It means that the system equations are dependent.
Infinitely many solutions.
Straight lines coincide.

See the lessons in this site
    - Solving systems of linear equations in two unknowns using the Cramer's rule
    - Geometric interpretation of the linear system of two equations in two unknowns

By analysing the second system, I do not use D%5By%5D as the condition recommend,
because it is pedagogically wrong (bad) pattern.