# How do you write the standard form of a line given (-1,-5) (-4,-2)?

Apr 29, 2017

See the entire solution process below:

#### Explanation:

First, 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:

$m = \frac{\textcolor{red}{- 2} - \textcolor{b l u e}{- 5}}{\textcolor{red}{- 4} - \textcolor{b l u e}{- 1}} = \frac{\textcolor{red}{- 2} + \textcolor{b l u e}{5}}{\textcolor{red}{- 4} + \textcolor{b l u e}{1}} = \frac{3}{-} 3 = - 1$

Next, use the point-slope formula to write an equation for the line. 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.

Substituting the calculate slope and the values from the first point in the problem gives:

$\left(y - \textcolor{red}{- 5}\right) = \textcolor{b l u e}{- 1} \left(x - \textcolor{red}{- 1}\right)$

$\left(y + \textcolor{red}{5}\right) = \textcolor{b l u e}{- 1} \left(x + \textcolor{red}{1}\right)$

We can now transform this to the Standard Linear form. 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

$y + \textcolor{red}{5} = \left(\textcolor{b l u e}{- 1} \cdot x\right) + \left(\textcolor{b l u e}{- 1} \cdot \textcolor{red}{1}\right)$

$y + \textcolor{red}{5} = \textcolor{b l u e}{- 1 x} - 1$

$1 x + y + \textcolor{red}{5} - 5 = 1 x \textcolor{b l u e}{- 1 x} - 1 - 5$

$1 x + y + 0 = 0 - 6$

$\textcolor{red}{1} x + \textcolor{b l u e}{1} y = \textcolor{g r e e n}{- 6}$