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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Being able to calculate the slope of a line is a very useful skill to have. Why? Watch this video lesson to find out and learn how to find the slope when working with various equations and various lines.

The **slope of a line** is how steep the line is and whether the line is an uphill or a downhill. Slopes can be whole numbers or fractions and either positive or negative. All slopes can be converted to a fraction form that tells you how much the line changes in the *y* direction over how much the line changes in the *x* direction. This is easy to remember as the phrase 'rise over run.'

For example, let's take a look at the slope 1/2. This slope is in fraction form. The top number tells you how the line changes in the *y* direction, or how much it rises. The number is a 1 and it is positive, so that tells us that the line goes up, or rises, by 1. If it was negative, then our line would go down by 1. The denominator is a 2, which means the line goes in the *x* direction or runs to the right for 2 spots. If we had a whole number, such as 3, we can convert to a fraction by remember that all whole numbers have a 1 in the denominator. So our 3 becomes a 3/1, which tells us that the line goes up 3 spots in the *y* direction and to the right 1 spot in the *x* direction.

Why do we need to know this information? Since equations for lines tell us about various situations and how much growth or decline there, being able to find the slope allows us to learn about how fast or slow the growth or decline is. Equations of lines come in different forms and from these equations we can find the slope directly by looking at the equations.

One type of equation that we come across is the point-slope form.

The *y* sub 1 and the *x* sub 1 stand for the coordinates of the point, and the *m* stands for the slope. Because the *m* stands for the slope, we can find the slope directly from this form of equation. For example, if we are given the equation *y* - 3 = 4(*x* - 2), we can see right away that the slope is 4 because the 4 is in the location of the *m*.

The other equation form of a line is the slope-intercept form.

In this form, the *m* stands for the slope and the *b* stands for the y-intercept, where the line crosses the y-axis. Again, because the equation has the *m* for the slope, we can see directly what the slope is. For example, if we have the equation y = -3/4*x* + 5, we look for where the *m* would be, and we see that there is a -3/4 in its place. That tells us that the slope is -3/4.

Finding the slope from these two equation forms is pretty straightforward. We can find the slope directly by looking at the equations. Finding the slope of a parallel line is also straightforward, but finding the slope of a perpendicular line requires a bit of manipulation.

A parallel line is a line with the same slope but in a different location. Because the definition of a parallel line calls for the same slope, we know right away that a parallel line will have the same slope. So if our line has a slope of 3/2, then our parallel line will also have a slope of 3/2. The actual equation may be different, but the slope is the same.

The slope of a perpendicular line is a bit different. Although different, it is not hard to calculate. All you have to do is to place your current slope underneath a -1. In other words, divide a -1 by the slope to get the perpendicular slope. This is called the *negative reciprocal*. You can check to see if you did it correctly by multiplying your current slope with the perpendicular slope. Multiplied together, it should equal -1.

For example, if our current slope is 5, then our perpendicular slope is -1/5. We have placed our current slope underneath a -1; we are dividing a -1 by our current slope. To check to see if we did it right, we multiply the 5 with the -1/5. What do we get? We get 5*(-1/5) = -1. Check. That is what we need, so we did it right.

What have we learned? We've learned that the **slope of a line** tells us how steep the line is and whether the line is an uphill or a downhill. To remember the slope, think of the phrase 'rise over run,' which tells us that in fraction form, the top number gives us how much change the line has in the *y* direction, while the bottom number gives us the change in the *x* direction.

For both the point-slope form and the slope-intercept form of equations, we can directly see what the slope is because both have an *m* in their equations that tells us the slope. All we have to look for is what number is in the *m* spot. For example, the equation *y* - 3 = 2(*x* - 1) is in the point-slope form and the slope is 2 since the 2 is in the *m* spot. The equation *y* = 3*x* + 4 is in the slope-intercept form with a slope of 3 since the 3 is in the *m* spot. For parallel lines, the slope is the same. For perpendicular lines, the slope is the negative reciprocal, which we find by dividing a -1 with our current slope.

By the end of this lesson you should be able to:

- Define slope
- Implement the point-slope and slope-intercept equations to solve for the slope of a line
- Calculate the slope of a parallel line and a perpendicular line

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Geometry: High School15 chapters | 160 lessons

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