The Equation of a Line

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A few days ago, I wrote about the skills the GMAT really tests within the category of coordinate geometry. The basic formulas won't get you very far--understanding and applying them in new situations is what will earn you a 600+ or 700+ score.

One thing I mentioned in the category of "basic formulas" is the equation of a line. You may recall it from high school geometry if you haven't already reviewed it for the GMAT. Most people learn the "slope-intercept" form:

y = mx + b

"m" stands for the slope, while "b" is the y-intercept--the point at which x = 0.

Deeper Understanding

That's all well and good, but how well do you really understand that concept? For instance, do you know the answers to the following questions:

  • How do you determine whether a certain point is on that line?
  • What does it mean if "m" is a positive fraction?
  • What if "m" is negative?
  • What happens if you increase "b"? Decrease "b"?
  • How can you handle equations that aren't given in slope-intercept form?

If you don't know the answers to the first four of those questions, I'll leave you to figure those out on your own. (Try coming up with a few sample equations, graphing them, and seeing how the values of "m" and "b" affect the location and angle of the line on the coordinate plane.)

The Other Equation of a Line

Often, the GMAT will test your ability to handle the last of those questions. What if, instead of y = 2x + 3, you're given x = (y-3)/2 ?

The high-school geometry way to handle that is to do some algebra and translate it back to slope-intercept form. Sometimes, that's fine on the GMAT, too, especially if you're only dealing with one such equation. What if, however, all five answer choices are given in this format?

The form "x = cy + d" doesn't tell you very much about the line--without translation, you don't get the slope or an intercept. But it does allow you to graph the line or determine whether a specific point is one the line.

For instance, if you wanted to find out whether the point (2, 6) was on the line y = 2x + 3, you could plug in 2 for x, and find that, when x = 2, y = 7. If (2, 7) is on the line, (2, 6) is not. You can run the same test even if the line isn't in slope-intercept form.

Given the equation x = (y-3)/2, plug in 6 for y, and find that, when y = 6, x = 3/2. So if (3/2, 6) is on the line, (2, 6) is not. Using the same technique, you could sketch the graph of the line using any form of the equation.

The slope-intercept form is a convenience, not a requirement. The GMAT is as much a time-management test as a skills test, so if you can handle something without going through every last textbook-approved step, you should do it.



About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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