# How do you write the equation in point slope form given (2, 5) (3,10)?

Jun 2, 2016

$y - 5 = 5 \left(x - 2\right)$

#### Explanation:

In general, given two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$
the slope can be calculated as
$\textcolor{w h i t e}{\text{XXX}} m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

and the equation of the line through these points using the point slope form is
$\textcolor{w h i t e}{\text{XXX")(y-y_1)=m(x-x_1)color(white)("XXXXX}}$see below for other forms

Given
$\textcolor{w h i t e}{\text{XXX}} \left({x}_{1} , {y}_{1}\right) = \left(2 , 5\right)$ and
$\textcolor{w h i t e}{\text{XXX}} \left({x}_{2} , {y}_{2}\right) = \left(3 , 10\right)$

$m = \frac{10 - 5}{3 - 2} = 5$

and the slope point form of the equation is
$\textcolor{w h i t e}{\text{XXX}} y - 5 = 5 \left(x - 2\right)$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The "slope-point form" may also appear in the form:
$\textcolor{w h i t e}{\text{XXX}} y - {y}_{2} = m \left(x - {x}_{2}\right)$
or
$\textcolor{w h i t e}{\text{XXX}} \frac{y - {y}_{1}}{x - {x}_{1}} = m$
or
$\textcolor{w h i t e}{\text{XXX}} \frac{y - {y}_{2}}{x - {x}_{2}} = m$
All these forms are equivalent.

Jun 2, 2016

$y - 5 = 5 \left(x - 2\right)$

graph{y=5x-5 [-10, 10, -5, 5]}

#### Explanation:

$y - {y}_{1} = m \left(x - {x}_{1}\right)$
Were ${y}_{1} \mathmr{and} {x}_{1}$ are points in which the line goes through, with ${x}_{1}$ being the x position and ${y}_{1}$ the y position which the points go through . Obviously the line goes through two of these points but lets just use $\left(2 , 5\right)$ as the point.
Next, we need the gradient which is $\frac{r i s e}{r u n}$. The rise between the two points are 5, with the run being 1. therefore, the gradient is $5$