# How do you find a standard form equation for the line with (4,1), (-3,6)?

Sep 6, 2017

See a solution process below: $\textcolor{red}{5} x + \textcolor{b l u e}{7} y = \textcolor{g r e e n}{27}$

#### Explanation:

First, we need to determine the slope of the line. 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 points in the problem gives:

$m = \frac{\textcolor{red}{6} - \textcolor{b l u e}{1}}{\textcolor{red}{- 3} - \textcolor{b l u e}{4}} = \frac{5}{-} 7 = - \frac{5}{7}$

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

The slope of an equation in standard form is: $m = - \frac{\textcolor{red}{A}}{\textcolor{b l u e}{B}}$

We can substitute the values from the slope for $\textcolor{red}{A}$ and $\textcolor{b l u e}{B}$ giving:

$\textcolor{red}{5} x + \textcolor{b l u e}{7} y = \textcolor{g r e e n}{C}$

We can substitute the values from one of the points in the problem for $x$ and $y$ to calculate $\textcolor{g r e e n}{C}$:

$\left(\textcolor{red}{5} \times 4\right) + \left(\textcolor{b l u e}{7} \times 1\right) = \textcolor{g r e e n}{C}$

$20 + 7 = \textcolor{g r e e n}{C}$

$27 = \textcolor{g r e e n}{C}$

Substituting this back into our equation gives:

$\textcolor{red}{5} x + \textcolor{b l u e}{7} y = \textcolor{g r e e n}{27}$