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Calculus: Help and Review13 chapters | 148 lessons

Instructor:
*Miriam Snare*

Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.

In this lesson, you will learn the definition and formula for writing the equation of a line in point-slope form. Then, we will look at a couple of examples. At the end, you will be able to test your knowledge with a quiz.

Using **point-slope form** means that you're supposed to write the equation of a line from knowing its slope and any point on the line. To visualize what's happening in this kind of a problem, let's imagine throwing a tennis ball onto a roof. The location where the ball hits the roof is the *point* you will use in the calculation. The steepness of the roof is the **slope**. The point-slope formula gives you an equation that describes the straight path that the tennis ball rolls down the roof.

The formula to find the equation of a line in point-slope form is:

To use this formula, you will substitute the coordinates of the known point for the *x sub 1* and the *y sub 1*. You will also replace the *m* with the slope that you know. The *x* and *y* without the subscript 1 will remain variables in the formula. Usually, after substituting the values, the equation is written in **slope-intercept form** which is *y* = *mx* + *b*. To get from point-slope to slope-intercept form, you need to distribute and combine like terms (i.e. the same variables) so that only the *y* is on the left side of the equation. Let's look at some examples of how to do that.

Example #1: Find an equation of the line with a slope of 3 that passes through the point (2, 4).

The *slope of 3* tells us to replace the *m* with 3. *The point (2, 4)* tells us that *x sub 1* will be replaced with 2 and *y sub 1* will be replaced with 4. Below you see the point-slope formula and below it, is the formula with the values filled in:

After this, you usually put the equation into *slope-intercept form* by solving the equation for *y*. So, we distribute the 3 into the parentheses to get *y* - 4 = 3*x* - 6. Then, we have to add 4 to both sides of the equation to get *y* by itself. So, the final equation is *y* = 3x - 2.

Let's try another example.

Example #2: A line passes through the point (3, -5) and has a slope of 2/3. Find an equation of this line.

Again, we will go through filling in the *point-slope formula* with the known values. This time, we need to be careful with that -5. Don't forget the negative! *The point (3, -5)* tells us that *x sub 1* will be replaced with 3 and *y sub 1* will be replaced with -5. Since we are subtracting -5 from *y*, we can write *y* + 5 instead of *y* - (-5). The *slope of 2/3* tells us to replace the *m* with 2/3. Below you see the point-slope formula and below it, is the formula with the values filled in:

From this step, we need to get the equation into *slope-intercept form* by solving it for *y*. So, we distribute the 2/3 into the parentheses to get *y* + 5 = (2/3)*x* - 2. Then, we have to subtract 5 from both sides of the equation. The final equation is *y* = (2/3)*x* - 7.

To write the equation of a line in **point-slope form**, you use the formula:

In that formula, the *x sub 1* and the *y sub 1* are known coordinates of a point on the line. The *m* is the slope of the line. After substituting the values, the equation is solved for *y*.

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Calculus: Help and Review13 chapters | 148 lessons

- What is a Function: Basics and Key Terms 7:57
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