**Point Slope Form**

### What is a Slope?

A slope is the measure of how steep a line is. It is usually represented as m, where m represents the rise over the run. (http://mathworld.wolfram.com/SlopeOfAPie… )

The point-slope form is an equation used in finding slopes of lines that can be easily found using point and slope form equations

Example: Find the point slope form for 2x+3y=6

First we find point and slope which are respectively (2,-1) and (-3,5) Now plug these points into point-slope form with “a” being equal to -1; “(x-(-1))/(-3)” now simplify this equation to get x=4/3

– point slope form is m=-1/2(x-(a))/(y-(b)); point slope form gives the point of intersection and the slope of a line. (http://mathforum.org/library/drmath/view… )

This point slope form can be used in graphing lines using these point and point-slope graph forms to give you the x-intercepts, y- intercepts, slopes, and point of intersection. These are important features to use when graphing a line since it determines whether your graph will be consistent with the point-slope equation or not. The point and point-slope graph forms are found below:

Point Formula: Point=(x-a)(y-b)

Point Slope Graph Form: (y- point =( point – slope ))/(x- point ) Now that I have told you all of this, take these new found skills and put them to the test by making a point and point slope graph. Good luck with this task, young grasshopper!

The point-slope form of a line is the point (x, y) and a slope m that satisfies the equation:

### What does it mean?

This point slope formula means geometrically that in order to find the equation of a line given two points on that line, we simply draw a straight line between those two points and read off the slope from that straight line. For example, if we have these two points (2, -3) and (-1,-5), then our point slope equation will be 5/2 minus 1/5 or 2/(2+(-1)) which just simplified to 3 . This shows us what our point slope formula says mathematically. Here is an example graph with a point slope equation and point slope formula for reference:

This point-slope method can be derived from the point-slope form equation as follows:

Which can also be written algebraically as:

And in point slope form, this is our point slope formula. This derivation shows us that this point slope form isn’t really a point slope formula at all but just uses the point-slope formula to find the edge points of a line. Remember that we must have two (or more) real numbers to work with ? Don’t forget! Also note that this particular point slope formula only works if our line passes through one of our given endpoints or comes very close to it. The closer the line gets to either endpoint when passing though it, the better.

where x is point (2, -3), y is point (-1,-5) and m is slope 3 . Now let’s take a look at how to find slopes using point slope form. To do this, we’ll first need to know what point-slope form looks like:

or in point slope form which would be written as:

Because finding slopes of lines requires division , we now multiply both sides of the equation by “m” instead and divide both sides by “m”. This gives us one final point-slope formula that is more helpful when working with slopes of lines:

Where “x” and “y” are points on the line and m is its slope. Now solve for m to find the point-slope form of the line:

### How to find the equation of a line with slope and coordinates of a point?

Let’s have a look at two exercises, to help understand the point-slope form equation better.

Identify point coordinates: x1 = 2, y1 = -3

Identify slope: m=2

Input point coordinates into point-slope form formula: y-y1=m(x-x1)

Simplify point slope formula and we get general equation of line: point slope form of equation y=-2x+3

To find point slope form, we have to:

– Calculate point slope formula.

– Find point x coordinates and point y coordinates.

Once we have the information in point slope form, we can then easily solve for m (slope of the line) by using point slope formula calculator which is available right here on this website!

Example 2: Let’s say that the point where a line intersects another line has coordinates (-8, 3). What is point slope form of this line?

Solution: First Step: find point slope formula: y – y1 = m (x – x1)

Second step: Substitute our values into point slope formula.

The coordinate of first point is (-8) and second one is 3. y – (-8) = m (x – 3) Now substitute point slope formula into point slope form. Point Slope Form: y = mx + b Substitute point slope formula into point slope form: y = m(x + 3) Solve point slope equation to obtain the value of the point-slope form . We get: y = 2x + 5 To verify, we can use point slope calculator.

Output : The point where a line intersects another line has coordinates (-8,3). What is point slope form of this line? Solution : Let’s find point coordinate x and point coordinate y using point-slope equation.

First step: Find point coordinates (using point-slope equation): x – point1 = point2 – point1 y + point1 = point2 – point1

Second step: Substitute our values into point slope formula. The coordinate of first point is (-8) and second one is 3. x – (-8) = 3 (as it’s a point which is not an endpoint). x – (-8) / 3 (we substitute the value of m from point slope form calculator in place “3”) So we get: 2x + y + 10 = 0 (point-slope equation without constants, i.e. variables are already solved for.)

Solve point slope equation to obtain the value of the point-slope form . We get: 2x + 5 = 0 Now substitute point slope

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