No solutions, one solution, infinitely many solutions?

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frctl

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Under what conditions on a and b will the following linear system have no solutions, one solution, infinitely many solutions?

2x - 3y = a
4x - 6y = b
 
What thoughts do you have about it? What would you look for to answer a question like this, if I gave you specific values for a and b?

One thing to consider is what the slopes of the two lines will be.
 
It appears the second eqn is twice the first.
I would look to see if a = b, if a > b or if a < b.
 
frctl said:
It appears the second eqn is twice the first.
I would look to see if a = b, if a > b or if a < b.
Click to expand...
Yes, the left-hand side of the second is twice the first; so what would be true of the right-hand side if they are the same line?

To check your guess, try a specific case. What if I gave you these equations?

2x - 3y = 6​
4x - 6y = 6​

There a = b; what happens? Is that a special case?
 
If a = b then the lines would intercept and the right hand side would reflect this pattern also.
 
frctl said:
If a = b then the lines would intercept
Click to expand...

Are you sure about that? If \(a = b\), then the two lines are:

\(\displaystyle
2x - 3y = a \\
4x - 6y = a
\)

You claim these lines intercept. So let's see if that's true for, say, \(a = 7\). Try graphing the two lines and you'll see they don't intercept. But, okay, maybe I just picked I convenient value of \(a\) to force that to happen. Think carefully about the lines and how they're related to one another, and why that means they don't intersect. That should formulate a hypothesis as to whether the lines might intersect for some other value of \(a\).

More broadly, you can make a lot of headway on this problem by thinking in generalities about any two arbitrary lines. Can you see why the system having no solutions is the same saying the lines don't intersect? And two lines never intersect if they're... (what?) Likewise, the system having infinitely many solutions is the same as saying the lines intersect infinitely many times. Two lines intersect at infinitely many points if they're... (what?) Finally, consider that if the lines intersect, they either intersect at infinitely many points or exactly one point. Why is this? Why is it not possible for the lines to intersect at two, or three, or even four points?
 
frctl said:
If a = b then the lines would intercept and the right hand side would reflect this pattern also.
Click to expand...
You didn't follow my suggestion at all, did you? Try actually solving my specific system of equations, in which a = b, and you'll see that there is no solution. You're jumping to conclusions without thinking carefully, which doesn't work well in math.
 
No solution means the lines will be parallel.
Infinite solutions means the lines are the exact same lines.
If they're not the same line and are not parallel then they must intersect in one point.

So under what condition for a and b, will each of those three cases hold?
 
No solution means the lines will be parallel.
a ≠ b

Infinite solutions means the lines are the exact same lines.
a = b

If they're not the same line and are not parallel then they must intersect in one point.
I can't find anything for these two equations.
 
Well, you're getting closer, but again I ask "Are you sure about that?" You're correct that there will be no solutions if the lines are parallel, and that infinitely many solutions if the lines are actually the same line, but you've got the conditions wrong. As I suggested last time, try a specific value, like, say, \(a = 7\) and graph the lines. Are they the same line?

It may help you figure out the proper conditions for the lines to be the same line if you think in terms of the equations. I'll give three examples of a pair of lines. Exactly one of these pairs is actually the same line. Can you identify which one? Further, think about what differentiates it from the two other examples, and see if you can figure out why that means the lines are the same.

\(\displaystyle
x + y = 5 \text{ and } 3x + 4y = 7 \\
4x + 4y = 20 \text{ and } 2x + 2y = 10 \\
2x + 6y = 17 \text{ and } x + 3y = 17
\)
As for when the lines intersect in exactly one point, have you considered that the answer might be it's impossible? ;)
 
Yes, 2x+6y=17 and x+3y=17 is the same line because a and b are the same.

Which conditions have I gotten wrong?
 
Please check what you are saying. Try graphing those two lines! They are not the same.

What do you have to do to the first equation to obtain the second?
 
frctl said:
Yes, 2x+6y=17 and x+3y=17 is the same line because a and b are the same.

Which conditions have I gotten wrong?
Click to expand...
NO they aren't the same line. For example the point (11,2) lies on the second line but not the first - so they can't be the same.

Try writing the lines in y = mx + c form.

The first one:
\(\displaystyle 2x - 3y = a\)
\(\displaystyle 3y = 2x - a\)
\(\displaystyle y=\frac{2}{3}x - \frac{a}{3}\)

Do the same for the second line. Compare the gradients and the y-intercepts.

When will these equations represent the same line?
When will they represent parallel lines?
When will they represent intersecting lines?
 
frctl said:
Yes, 2x+6y=17 and x+3y=17 is the same line because a and b are the same.
Click to expand...

Nope, that's not correct. Did you try graphing the lines, as I've suggested at least three times now? I'm starting to feel like you're just guessing or going on what feels right, without any regard for the truth of the matter. It's somewhat unfortunate, but many times in mathematics what seems intuitive or "obvious" is actually wrong.
 
Draw these two lines:
Y= 2x+3
Y=2x+5
Are they the same?
Now lets rearrange them
Y-2x=3
Y-2x=5
The coefficient of Y and X is the same on both above equations but that doesn’t mean they are the same... those coefficient will give our two lines the same slope but they are not the same line
 
2x - 3y = a ......(1)
4x - 6y = b ......(2)

(1) 3y = 2x - a
y = (2/3)x - (a/3)
(2) 6y = 4x - b
y = (4/6)x - (b/6)
y = (2/3)x - (b/6)

The coefficient of Y and X is the same on both above equations
This means they have the same slope but do not intercept.

When will these equations represent the same line?
...
When will they represent parallel lines?
...
When will they represent intersecting lines?
...
 
When two lines have the same slope, under what conditions will they be the same line?

You're learning this; so in order to learn it well, you need to state the conclusion, rather than someone else. Give it another try.
 
frctl said:
Yes, 2x+6y=17 and x+3y=17 is the same line because a and b are the same.

Which conditions have I gotten wrong?
Click to expand...

lets make it easier to spot
2x+6y=17
x+3y=17

let multiply the second one by two so they look exactly the same
2x+6y=17
2(x+3y)=2*17 --- > 2x+6y=34

they are not the same equation , are they?
 
Yes they are, this means they will be the same when a < b or b > a.
 
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