# How do you write an equation in standard form given point (-1,6) and (3,-2)?

Feb 4, 2017

$\textcolor{red}{2} x + \textcolor{b l u e}{1} y = \textcolor{g r e e n}{4}$

#### Explanation:

We will need to first find an equation in point-slope form. Which, we must first determine the slope. The slope can be found by using the formula: $m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Where $m$ is the slope and ($\textcolor{b l u e}{{x}_{1} , {y}_{1}}$) and ($\textcolor{red}{{x}_{2} , {y}_{2}}$) are the two points on the line.

Substituting the values from the two points in the problem gives:

$m = \frac{\textcolor{red}{- 2} - \textcolor{b l u e}{6}}{\textcolor{red}{3} - \textcolor{b l u e}{- 1}}$

$m = \frac{\textcolor{red}{- 2} - \textcolor{b l u e}{6}}{\textcolor{red}{3} + \textcolor{b l u e}{1}}$

$m = - \frac{8}{4} = - 2$

The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

We can substitute the slope we calculated and the first point giving:

$\left(y - \textcolor{red}{6}\right) = \textcolor{b l u e}{- 2} \left(x - \textcolor{red}{- 1}\right)$

$\left(y - \textcolor{red}{6}\right) = \textcolor{b l u e}{- 2} \left(x + \textcolor{red}{1}\right)$

The standard form of a linear equation is: $\textcolor{red}{A} x + \textcolor{b l u e}{B} y = \textcolor{g r e e n}{C}$

where, if at all possible, $\textcolor{red}{A}$, $\textcolor{b l u e}{B}$, and $\textcolor{g r e e n}{C}$are integers, and A is non-negative, and, A, B, and C have no common factors other than 1

We can now transform our point-slope form to this form:

$y - \textcolor{red}{6} = \left(\textcolor{b l u e}{- 2} \times x\right) + \left(\textcolor{b l u e}{- 2} \times \textcolor{red}{1}\right)$

$y - 6 = - 2 x - 2$

$2 x + y - 6 + 6 = 2 x - 2 x - 2 + 6$

$2 x + y - 0 = 0 + 4$

$2 x + y = 4$