Question

What is the equation in point-slope form of the line through the points (10,3) and (12,11)

Answer 1

Using the first point (10,3) the point-slope form gives:
`y-3=4(x-10)`

Using the second point (12,11) the point-slope form gives:
`y-11=4(x-12)`

Explanation:

Notice that you are given two points
`(x_1,y_1)` and
`(x_2,y_2)` on the plane through which the line has to go through.

We can start then by finding the value of the slope for a segment that joins such points via the equation for the slope:
`slope=(y_2-y_1)/(x_2-x_1)`. Which in our case, if we call
`(x_1,y_1)` = (10,3) and
`(x_2,y_2)` = (12,11) give us:

`slope=(y_2-y_1)/(x_2-x_1)\\slope=(11-3)/(12-10)\\slope=(8)/(2)\\slope=4`

Now that we have the slope of the line, we can write the "point-slope" form of it by using the information of on the general form of a line of slope "m" going through the point
`(x_0,y_0)` in point-slope form:

`y-y_0=m(x-x_0)`

we know our slope must be "4", and we can use any of the given points (for example (10,3) as the specific point
`(x_0,y_0)`, resulting in:

`y-y_0=m(x-x_0)\\y-3=4(x-10)`

Of course, we could have used the other point as well, which would give us the following:

`y-y_0=m(x-x_0)\\y-11=4(x-12)`

and although they look like different equations, they basically represent the very same equation, fact that we can verify by solving for "y" in both expressions:

`y-3=4(x-10)\\y-3=4x-40\\y=4x-40+3\\y=4x-37`

`y-11=4(x-12)\\y-11=4x-48\\y=4x-48+11\\y=4x-37`