Solve the system by the method of your choice. Identify systems with no solutions and systems with infinitely many solutions, using set notation to express their solutions:
(0 ,-5)
(0 , 0)
(-2 ,-1)
(-2,-3)
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To solve the given system of equations, we'll consider each pair of equations and look for a pattern or relationship between them.
First, let's write out the system:
(0, -5)
(0, 0)
(-2, -1)
(-2, -3)
Let's pair the equations (1) and (2) and (3) and (4):
Equations (1) and (2):
0 = 0
-5 = 0
For equations (1) and (2), we see that both equations have the same x-coordinate but different y-coordinates. This indicates that the system has no solution because the points (0, -5) and (0, 0) do not intersect (they are parallel lines). Thus, we can represent this system as:
{(0, -5), (0, 0)}
Equations (3) and (4):
-2 = -2
-1 = -3
For equations (3) and (4), we observe that both equations have the same x-coordinate and the same y-coordinate. This implies that the two equations represent the same line and the system has infinitely many solutions. Hence, we can represent this system as:
{(-2, -1)} or {(-2, -3)}
To summarize:
System with no solution: {(0, -5), (0, 0)}
System with infinitely many solutions: {(-2, -1)} or {(-2, -3)}