Sometimes, the GMAT give you the line in another form (e.g. Sometimes the GMAT will give you the equation of a line already in y = mx + b form. That is a powerful and often underappreciated mathematical idea. This is a very deep idea - x & y don’t equal any one pair of values rather, every single point (x, y) on the line, the entire continuous infinity of points that make up that line - every single one of them satisfies the equation of the line. This is not the “x” of ordinary solve-for-x algebra. By contrast, x & y (sometimes call the “graphing variables”) do not equal just one thing. For any given line, m & b are constants: for a given line, both m & b equal a fixed number. The b is the y-intercept: where the line crosses the y-axis. There are a few different ways to write a line, but the most popular and easiest to understand is y = mx + b. That is a very deep idea.Ī straight line is a very simple picture, and not surprisingly it has a very simple equation. In more practical terms, every equation (an algebraic object) corresponds to a picture (a geometric object). It allowed for the unification of two ancient branches of mathematics: algebra and geometry. Although you may have met this sometime in middle school math and may now take it for granted, it is actually a brilliant mathematical device. The technical name of the x-y plane is the Cartesian plane, named after its inventor, Mr. Let’s get a bit philosophical for a moment. Again, I highly recommend performing this visual check every time you calculate slope. Even a rough sketch would verify that, yes, the slope should be negative. Your sketch, of course, does not need to be this precise. Here’s a sketch of this particular calculation: Whenever you find a slope, I strongly suggest doing a rough sketch, just to verify that the sign of the slope (positive or negative) and the value of the slope are approximately correct. Slope is definitely something you need to understand for the GMAT Quantitative section. Now, rise/run = –3/7 - that’s the slope. For the sake of argument, we’ll say that’s the order - (–2, 4) is the “first” and (5, 1) is the “second.” The rise is the change in height, the change in y-coordinate: 1 – 4 = –3 (notice, we had to do second minus first, which gave us a negative here!) The run is the horizontal change, the change in x-coordinate: 5 – (–2) = 5 + 2 = 7 (remember: subtracting a negative is the same as adding a positive!). Once we have rise & run, divide them, rise divided by run, to find the slope.įor example, suppose our points are (–2, 4) and (5, 1). The run is the horizontal change - the change in the x-coordinate (again, second minus first). The rise is the vertical change - the change in y-coordinate (second point minus first). It actually doesn’t matter which one we say is the first and which one, the second: all that matters is that we are consistent. To calculate rise and run, first have to put the two points in order. There is very algebraic formula for the slope, and if you know that, that’s great! If you don’t know that formula, or used to know it and can’t remember it, I will say: fuhgeddaboudit! Here’s a much better way of thinking about slope. Slope is a measure of how steep a line is. If these problems make your head spin, you have found the right post. What is the value of b?ģ) A line that passes through (–1, –4) and (3, k) has a slope = k. Here are a set of practice GMAT questions about the Cartesian plane.ġ) What is the equation of the line that goes through (–2, 3) and (5, –4)?Ģ) The line y = 5x/3 + b goes through the point (7, –1).
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