SOLUTION: In each of the following systems of equations, look at the system and determine whether it has infinitely many solutions, no solution, or one solution. Explain in each case how you

Algebra ->  College  -> Linear Algebra -> SOLUTION: In each of the following systems of equations, look at the system and determine whether it has infinitely many solutions, no solution, or one solution. Explain in each case how you      Log On


   

Question 997848: In each of the following systems of equations, look at the system and determine whether it has infinitely many solutions, no solution, or one solution. Explain in each case how you came to your conclusion. You should not attempt to solve any of the systems.
a) Let k represent a Real Number.
y = 2x + k
y = 2x – k
b) Let a represent a Real Number.
y = -2x + a
y = 12 x – a
Answer by ikleyn(52781) About Me  (Show Source):
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a) Let k represent a Real Number.
y = 2x + k
y = 2x – k
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These equations represent parallel straight lines.
These lines coincide if  k=0.  At this value of  k  the system has infinitely many solutions.

If  k=/=0,  then these lines are parallel and do not coincide.  The system has no solutions in this case.


b) Let a represent a Real Number.
y = -2x + a
y = 12 x – a.

These straight lines have different slopes.  They are not parallel,  hence,  they have an intersection.  It means that the system has an unique solution.