One of the most important properties of a straight line is in how it angles away from the horizontal. This concept is reflected in something called the "slope" of the line.

Let's take a look at the straight line y = ( ^{2}/_{3} ) x– 4.

Pick two x's and solve for each corresponding y: If, say, x = 3, then y = ( ^{2}/_{3} )(3) – 4 = 2 – 4 = –2. If, say, x = 9, then y = ( ^{2}/_{3} )(9) – 4 = 6 – 4 = 2. (By the way, I picked the x-values to be multiples of three because of the fraction. It's not a rule that you have to do that, but it's a helpful technique.) So the two points (3, –2) and (9, 2) are on the line y = ( ^{2}/_{3} )x – 4.

To find the slope, you use the following formula:

he subscripts merely indicate that you have a "first" point (whose coordinates are subscripted with a "1") and a "second" point (whose coordinates are subscripted with a "2"); that is, the subscripts indicate nothing more than the fact that you have two points to work with. It is entirely up to you which point you label as "first" and which you label as "second". For computing slopes with the slope formula, the important thing is that you subtract the x's and y's in the same order. For our two points, if we choose (3, –2) to be the "first" point, then we get the following:

y-value above, the –2, was taken from the point (3, –2) ; the second y-value, the 2, came from the point (9, 2); the x-values 3 and 9 were taken from the two pointsin the same order. If we had taken the coordinates from the points in the opposite order, the result would have been exactly the same value:

x-values in the same order as you subtracted the y-values. Because of this, the slope formula can be written as it is above, or alternatively it can be written as:

x-values in the same order as you had subtracted your y-values.

May 10th, 2014

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