Question

How do you write the point slope form of the equation given (-5,-1) and (0,-5)?

Answer 1

`y+1=-4/5(x+5)color(white)("xxx")orcolor(white)("xxx")y+5=-4/5(x-0)`

Explanation:

Note that the slope between the points `(-5,-1)` and `(0,-5)` is
`color(white)("XXX")color(green)m=(Deltay)/(Deltax)=(-1-(-5))/(-5-0)=color(green)(-4/5)`

The general slope-point form for a line with slope `color(green)m` through the point `(color(blue)(hatx),color(red)(haty))` is
`color(white)("XXX")y-color(red)(haty)=color(green)m(x-color(blue)(hatx))`

We can use either of the given points for our `(color(blue)(hatx),color(red)(haty))`.
If, for example, we use `(color(blue)(hatx),color(red)(haty))=(color(blue)(-5),color(red)(-1))`
then our slope-point form becomes
`color(white)("XXX")ycolor(red)(+1)=color(green)(-4/5)(xcolor(blue)(+5))`
[using `(color(blue)(hatx),color(red)(haty))=(color(blue)(0),color(red)(-5))` gives the alternate, but equivalent, version shown in the Answer].

Answer 2

The point-slope form is `y+1=-4/5(x+5)`.

Explanation:

Slope

You first need to determine the slope from the given points. The formula for determining slope is:

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

where:

`m` is the slope, `(x_1,y_1)` and `(x_2,y_2)` are the two points.

Plug the known values into the formula. I'm going to use `(-5,-1)` for Point 1 and `(0,-5)` for Point 2. It doesn't matter which point you make 1 or 2. The result will be the same.

`m=(-5-(-1))/(0-(-5))`

Simplify.

`m=(-5+1)/(0+5)`

Solve.

`m=-4/5`

Point-slope form

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

where:

`m=-4/5`, `(x_1,y_1)` is one of the points. It doesn't matter which one. I'm going to use `(-5,-1)`.

Plug in the known values.

`y-(-1)=-4/5(x-(-5))`

`y+1=-4/5(x+5)` `larr` Point-slope form

You can solve for `y` to convert it into the slope-intercept form:

`y=mx+b`,

where:

`m` is the slope, `-4/5`, and `b` is the y-intercept.

`y+1=-4/5(x+5)`

Expand the right-hand side.

`y+1=-4/5x-4`

Subtract `-1` from both sides.

`y+1=-4/5x-4-1`

Simplify.

`y=-4/5x-5` `larr` Slope-intercept form.