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Lesson 151: More About Slope
You're already learned a lot about slope, as is applies to linear equations. In this lesson, we'll learn some more facts about slope, and some new formulas.
Before starting this lesson, review all of the previous lessons on slope and linear equations. Make sure that you fully understand them. Recall that we defined slope as the change in y over the change in x. All of the alternate definitions that we learned are equivalent. It is important to understand the actual formula for slope:
m = (y2 - y1) / (x2 - x1). To use this formula, pick one point on the line to call Point 1, and another point to call Point 2. Once you have done this, do not get the two points confused, and do not change which one is which while in the middle of the problem. That is a common mistake. Let's try an easy example. Find the slope of the line that crosses through the points (2, 1) and (4, 7). I'll call the first point 1, and the second point 2. I could have done the opposite, as long as I was consistent. We'll do (7-1) / (4-2). We get 6/2, which is 3. Our slope, m, is 3.
Here is a trickier example. Find the slope of the line that
crosses through the points
(-2, -1) and (3, -5). Again, I'll call the points 1 and 2,
respectively. We must calculate
(-5 - -1) / (3 - -2). Notice how all I did was use the slope
formula, carefully substituting the appropriate values. Now it
is just a matter of being very careful with doing the signed number
arithmetic, remembering all of the rules, including how to subtract
a negative. We get -4/5 as the value of the slope. Make
sure you see how.
Very often we are only given a slope and a single point on the
line, and we are asked to write the equation of the line. One
way to do this is to plot the given point, and then use the slope to
help us figure out other points that are on the line. However,
that is time consuming. An easier way is to use a special
formula called the point-slope equation:
y - h = m(x - k). In this formula, m is the given
slope, k is the x-coordinate of the given point, and h is the
y-coordinate of the given point.
Let's try an example. What is the equation of a line with slope 3, that passes through the point (4, 5). Using our formula, we have y - 5 = 3(x - 4). Now we must use basic algebra to manipulate this equation into standard y=mx+b form. We have y - 5 = 3x - 12, which leads to y = 3x - 7. We now see that the y-intercept is -7. Notice that the point-slope equation involves subtraction. That means that you'll need to be very careful with your subtraction, if the given point involves negative numbers.
It's important to realize that if two lines have the same slope, then they are parallel. They slant at the exact same angle, so they will never meet, unless they happen to be the exact same line, and are on top of each other.
Also, memorize the fact that two perpendicular lines will have slopes that are negative reciprocals of each other. For example, if one has a slope of 2/3, the other will have a slope of -3/2. If one has a slope of 4, the other has a slope of -1/4.
As a review, be sure to memorize the fact that lines that slant up and to the right have positive slopes. Lines that slant up and to the left have negative slopes. Horizontal lines have a slope of 0, and vertical lines have undefined slopes.
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