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Find the equation of the line that goes through the points $(2, 9)$ and $(8, 6)$. Write the equation in standard form.

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What have you tried? What is your standard form? Can you just plug the given points into it? –  Ross Millikan Jun 29 '11 at 20:57
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Since you're new, you are probably unaware that many people consider it impolite to have posts that read as if you were giving orders or handing out assignments (in the imperative mode: "Find", "Solve", etc). The best way to get feedback is to provide some context for your question (why are you asking this question? in what context did you find it/was it assigned), and say why or where you are having difficulty. In addition, if the problem is homework or an assignment, many of us would prefer if you add the tag [homework] to the list of tags. Please edit your question accordingly. –  Arturo Magidin Jun 29 '11 at 21:00
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I suggest that you write down the standard form of a general line - which will be an equation. Then substitute each of the two points in turn to find the conditions for them to satisfy the equation, and use those conditions to work out the equation explicitly. Then see what happens with two general points and how you might solve that. See if you can understand what's going on (you may get a form which equates two expressions for the slope of the line, for example). You'll learn more by doing that than picking a solution off wikipedia. –  Mark Bennet Jun 29 '11 at 22:28

3 Answers 3

Mark Bennet's comment above is excellent. While we wait anxiously to hear about your trials and travails with the problem, I think I'll provide a few extra ways to attack this problem. It's always nice to know that there are options.

1: Buy some graph paper, a ruler, and a protractor. Plot the two points and connect them with the ruler, but be very precise. I recommend a finely pointed .3 mm lead pencil. Pay very particular attention to where it crosses the y-axis, and call this point $b$. Then use the protractor to measure the angle of the line with any horizontal grid marking (remember, it's graph paper!). The slope is given by $\tan \theta$, where $\theta$ is the angle you measured. That's all for that one! But sometimes the boxes on graph paper are so small, that I do the next one.

2: Buy post-it notes, a meter-stick, and a big protractor. Very carefully place the post-it notes in a grid-like pattern. The larger size will lead to larger accuracy. Repeat the steps in number 1 and you'll get your answer.

3: Finally, the general way. It's time to wash our hands of the dirty methods I mentioned above and go for something more ivory tower. Note that our two points that we know on the line are both lattice points, and so the slope and y-intercept will at least be rational points. But the rationals are countable! And the cross product of two countable sets is countable! So take an enumeration of the cross product of two countable sets and go down the list, testing to see if our two points fall on the line. Now you don't need to buy post-it notes, a ruler, or a protractor! And eventually, you'll get there!

If none of these work, there are some great statistical programming languages out there that can find the line of best fit for a set of data. Maybe putting these two points in the freely available statistical software R will help. The people over at CrossValidated could probably help with that, if necessary.

Keep us posted on your progress! And I just wanted to tell you good luck. We're all counting on you.

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The equation of the line is given by:

  • $y-y_{1} = m \cdot (x-x_{1})$, where $m$ is the slope which is given by $m= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$.
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