Question

How do you write the point slope form of the equation given (5,-6) and (2,3)?

Answer 1

See the solution process below:

Explanation:

First, determine the slope of the line. The slope can be found by using the formula: `m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))`

Where `m` is the slope and (`color(blue)(x_1, y_1)`) and (`color(red)(x_2, y_2)`) are the two points on the line.

Substituting the values from the points in the problem gives:

`m = (color(red)(3) - color(blue)(-6))/(color(red)(2) - color(blue)(5)) = (color(red)(3) + color(blue)(6))/(color(red)(2) - color(blue)(5)) = 9/-3 = -3`

The point-slope formula states: `(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))`

Where `color(blue)(m)` is the slope and `color(red)(((x_1, y_1)))` is a point the line passes through.

Substituting the slope we calculated and the values from the first point in the problem gives:

`(y - color(red)(-6)) = color(blue)(-3)(x - color(red)(5))`

Solution 1) `(y + color(red)(6)) = color(blue)(-3)(x - color(red)(5))`

We can also substitute the slope we calculated and the values from the second point in the problem giving:

Solution 2) `(y - color(red)(3)) = color(blue)(-3)(x - color(red)(2))`

Answer 2

`y=-3x+9`

Explanation:

First, find the slope between the points. You can find slope using:

`m=(y_2-y_1)/(x_2-x_1)`

Plug into the formula:

`m=(3-(-6))/(2-5)=9/-3=-3`

Now we have the slope. We can now plug into the point-slope form which is:

`y-y_1=m(x-x_1)`

It doesn't really matter which point we plug in so I'll plug in `(2,3)` since I don't like dealing with negative numbers:

`y-3=-3(x-2)`

Distribute the `-3`:

`y-3=-3x+6`

Add `3` to both sides:

`y=-3x+9`