Paired Equations: Unique, No, or Infinite Solutions?

[elfy]

Homework Statement

For each of the following equations, without attempting to solve them, determine whether there exists (1) a unique solution, (2) no solution or (3) an infinite # of solutions

Homework Equations

A) x+y =1 , x=y
B) y=x/2 - 1 , 2y = x-2
C) x+y = 1, x+y = 10

The Attempt at a Solution

I don't know how to solve this without trying to solve the equations. Are there some types of general guidelines in terms of the number of solutions? For instance in A) I would say there is 1 solution (x,y = 0,5) but that's just because I tried solving it.

 

[HallsofIvy]

Geometrically, two distinct lines either intersect or are parallel. If they intersect there is a unique point of intersection, if they are parallel, there is none. Which of those equations correspond to parallel lines? (Look at the slopes.)

Of course, if the two different equations really give the same line, then there are an infinite number of solutions- every point on that line. Which of those pairs of equation really just define one line?

 

[elfy]

First of all - Thank you for your time and help!

Trying to determine the number of solutions based on what you explained, I think that B) is the same line. Although I'm not sure if I cheated because although I did not solve anything, I just saw that if you solve the second equation wrt Y, you got the same equation as the first one. Is this allowed? :) Thus, B) has an infinite amount of solutions.

In C) we get 2 straight lines (if I have done this correctly) and they intersect once - so 1 solution for C).

Regarding A I also think that it's 2 straight lines which intersect at 0.5 - so a unique solution?

I hope that I haven't completely missunderstood?

Again thanks for your time!

 

[yenchin]

In C) we get 2 straight lines (if I have done this correctly) and they intersect once - so 1 solution for C).

Where do they intersect?

 

[Mark44]

In C) we get 2 straight lines (if I have done this correctly) and they intersect once - so 1 solution for C).

Yes, they are two straight lines, but why do you think they intersect? Are you just guessing?

 

[elfy]

Sorry, my bad!

The two lines run parallell to each other, intersecting Y at 1 and 10, thus there is no solution!
is that correct? :) It was not my intention to guess, I just got it wrong last night hehe.

However, it says "without attempting to solve the equations" but I need to draw the graphs in order to see what they look like. Is there some other way of doing it, as I kind of am solving them I suppose, or atleast doing something with them (sketching the graphs on a piece of paper) and I don't know if that's cheating or not :)

Thank you all for your help and patience with me! :)

 

[yenchin]

Sorry, my bad!

The two lines run parallell to each other, intersecting Y at 1 and 10, thus there is no solution!
is that correct? :) It was not my intention to guess, I just got it wrong last night hehe.

However, it says "without attempting to solve the equations" but I need to draw the graphs in order to see what they look like. Is there some other way of doing it, as I kind of am solving them I suppose, or atleast doing something with them (sketching the graphs on a piece of paper) and I don't know if that's cheating or not :)

Thank you all for your help and patience with me! :)

One way to tell that there is no solution is that the two equations are incompatible. You can't have x+y=10 and x+y=1 simultaneously because 1 is not equal to 10. :-)

 

[elfy]

Ahh that makes sense - Thanks! :)

So you are allowed to alter the equations (for instance in B) solving for Y, yields the same equation)?

I initially thought that you had to just look at them without doing anything, and just be able to see how many solutions there were hehe

 

[eumyang]

So you are allowed to alter the equations (for instance in B) solving for Y, yields the same equation)?

Sure, why not? I think in the Algebra 1 book that I use, it says that for these types of problems, rewrite the equations in slope-intercept form. Then it's pretty straightforward to see how many solutions a system has.

 

[elfy]

Thanks for clearing that up! :)

I really appreciate all the help and guidence you have given me! Thanks for your time and effort!