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$2x + 2y + 2 = 0$ and $7x + 7y + 7 = 0$. Are these equations consistent, inconsistent, or dependent?

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10  
And your thoughts on the matter are...? (That is: what have you tried? Where are you stuck or confused?) – Arturo Magidin Mar 14 '11 at 17:14
I need to know if they are consistent, inconsistent, or dependent... – Barny William Mar 14 '11 at 17:18
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Yes, you were very clear on what you need to know. What you haven't been clear on is on why you are unable to figure it out on your own. The purpose of this site is not for us to do your homework for you, but for you to get help. In order for us to give you appropriate help (so that you can solve these problems in the future) it is important (if not imperative) for us to know what it is that is giving you trouble. So... why are you unable to solve this problem? Where are you confused? What did you try that didn't work out? These are things you should tell us if you want to learn. – Arturo Magidin Mar 14 '11 at 17:20
also, in the future, please make your titles more descriptive. – yunone Mar 14 '11 at 17:21
The second equation is 7/2 times the first equation, so they are essentially the same equation. – quanta Mar 20 '11 at 16:37

4 Answers

up vote 1 down vote

(I'll try to spoon-feed a little less...)

First, we try to decide if these two equations have any solutions. In other words, is there (at least) one ordered pair (x, y) that is a solution (i.e. yields a true equality) for both equations?

I suggest that you use the "elimination" method for solving systems of equations.

Here's a hint: if you ever get an equation with no x's or y's that is always false, like 3 = 0, then the system is inconsistent.

Likewise, if you come to an equation that is always true, like 2 = 2 or 0 = 0, then the system is consistent and dependent.

Otherwise, you'll have one solution, which is called "independent and consistent".

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up vote 1 down vote

Since The Chaz has given hints in the direction of symbolic manipulation, I'll hint at a graphical way of thinking:

  • What are the slopes of the two lines? What does that tell you about the lines?
  • Can you find the $y$-intercepts of the lines? Or the coordinates of other point(s) on the lines?
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Said hints are... 17 pixels away, in case the link doesn't work! – The Chaz 2.0 Mar 14 '11 at 18:55
@The Chaz: :) That's true now, but the ordering of answers can change and sometimes people post more than one answer, so I tend to link things that may be obvious now on the off chance they are not obvious in the future. – Isaac Mar 14 '11 at 18:56
Ahah. So I need some sort of inline code that will measure the varying distance (in pixels)! But seriously... thanks, from a newbie. – The Chaz 2.0 Mar 14 '11 at 19:01
up vote 0 down vote

Both of your equations can be simplified to $y=-1x-1$.

What does this equation look like to you? You can graph it if you have a calculator or compare it to basic functions.

You should also know what properties the three words you provided hold, and The Chaz has explained them very clearly.

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up vote 0 down vote

HINT: The system of linear equations is indeed dependent.

But do you know the reason why this result occured?

What can we tell about how the solution would look like?

What did you get for an answer when trying to solve this if you solved it, or by inspection by

looking at the coefficients of the variables $x \text{ and } ~y$?

Then I can lead you to the reasoning of why the system is a dependent one.

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