Slope Formula

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Lesson Objective

This lesson shows you how the slope formula is derived and some examples on using the formula.

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About This Lesson

After defining the slope of a line as the ratio of the 'change in y' and 'change in x', we can use this definition to derive the slope formula.

This lesson will show you the steps to derive this formula. Also, you will able to see some examples on using this formula.

You should proceed by reading the study tips and watch the math video below. After that, you can try out the practice questions.

Slope formula or gradient formula

Study Tips

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Tip #1

Understand how the 'change in y' and 'change in x' are calculated. To recall them, you can watch the first math video in the slope of a line lesson.

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Tip #2

It is important to understand the slope formula before using it. You will be more comfortable with the formula once you have understood it.

Now, watch the following math video to learn more.

Math Video

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Click play to watch video

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Math Video Transcript

In this lesson, we will learn how to derive and use the slope formula.

From the previous lesson, we learn that the slope of a line is equals to, 'change in y' divides by 'change in x'.

Using this definition, we can now derive the slope formula.

Let's put the first point on the coordinate plane.

Now,since we are deriving a formula, we can represent the x-coordinate and y-coordinate as x1, and y1 respectively.

Let's put the second point on the plane.

Again, we can represent the x-coordinate and y-coordinate as x2, and y2 respectively.

Now, with these two points, we can draw a straight line and derive the slope formula from here.

Now, imagine that we run from, x1 to x2. The change in x would be x2 minus x1.

Alright, with this, we can replace 'change in x' with 'x2 minus x1'.

Next, referring to the y-coordinates, when we climb up from here, the "change in y" will be, y2 minus y1.

Now, with this, we can replace 'change in y' with 'y2 minus y1'.

Finally, the slope is equals to, "y2 minus y1" divides by "x2 minus x1".

Following the convention, we can represent the slope with the variable m.

So, we now have the slope formula, m equals to, "y2 minus y1" divides by "x2 minus x1".

Alright, let's use this formula to find the slope of this line.

Let's view the actual coordinates for the first point. we have, x1 as 2.0, and y1 as 3.0.

Similarly, for the second point, we have x2 as 5.0, and y2 as 6.0.

Now, we can find the slope of this line by simply substituting these coordinates into the slope formula.

Let me show you, substituting y2 with 6, y1 with 3, x2 with 5, and x1 with 2.

Now, we can see that we do not need these brackets. So, let's remove them.

Let's simplify this, negative multiply by bracket 3.0 gives negative 3.0.

Negative multiply by bracket 2.0 gives negative 2.0.

6.0 minus 3.0 gives positive 3.0. 5.0 minus 2.0 gives positive 3.0.

Now, positive 3 divides by positive 3 gives positive 1.

So, the slope of this line is positive 1.

Let's take a look at another example. But first, let me change the coordinates of these points.

Using this formula, we can substitute y2 with 4.0, y1 with negative 1.0, x2 with 1.0, and x1 with 6.0.

Now, we can see that we do not need these brackets. So, let's remove them.

Let's simplify this, negative multiply by bracket negative 1.0 gives positive 1.0

Negative multiply by bracket 6.0 gives negative 6.0.

4.0 plus 1.0 gives positive 5. 1.0 minus 6.0 gives negative 5.

Now, positive 5.0 divides by negative 5.0 gives negative 1.

So, the slope of this line is negative 1.

That's all for this lesson, try out the practice question to test your understanding.

Practice Questions & More

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Multiple Choice Questions (MCQ)

Now, let's try some MCQ questions to understand this lesson better.

You can start by going through the series of questions on slope formula or pick your choice of question below.

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