Question

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

Answer 1

Answer:

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: $y-11=4(x-12)$

Step-by-step 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: $slope=\frac{y_2-y_1}{x_2-x_1}$. Which in our case, if we call  $(x_1,y_1)$ = (10,3) and $(x_2,y_2)$  = (12,11) give us:

$slope=\frac{y_2-y_1}{x_2-x_1}\\slope=\frac{11-3}{12-10}\\slope=\frac{8}{2}\\slope=4$

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:

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

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