# How do you write an equation in standard form for the line through (1,3) and (-10,6)?

May 27, 2015

Start by determining the slope of the line through the given points;
then write the equation for the line in point-slope form;
finally convert to standard form.

Slope
The slope of a line is determined by the difference between 2 y-coordinate values divided by the difference between 2 corresponding x-coordinate values.
Given the points $\left(1 , 3\right)$ and $\left(- 10 , 6\right)$
Slope: $m = \frac{3 - 6}{1 - \left(- 10\right)} = - \frac{3}{11}$

Point-Slope Form
The point-slope form for a line with slope $m$ through a point $\left({y}_{1} , {x}_{1}\right)$ is
$\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)$
For the given values this becomes
$y - 3 = \left(- \frac{3}{11}\right) \left(x - 1\right)$

Standard Form for Linear Equation
The standard form for a linear equation is
$A x + B y = C$ (usually with the restriction that $A \ge 0 \mathmr{and} A \epsilon \mathbb{Z}$
$y - 3 = \left(- \frac{3}{11}\right) \left(x - 1\right)$

$y = - \frac{3}{11} x + \frac{3}{11} + 3$

$\frac{3}{11} x + y = \frac{36}{11}$

$3 x + 11 y = 36$

(...and our work here is done)