# How do you write the standard form of a line given (3, 2) and (5, 6)?

May 24, 2017

See a solution process below:

#### 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:

$m = \frac{\textcolor{red}{6} - \textcolor{b l u e}{2}}{\textcolor{red}{5} - \textcolor{b l u e}{3}} > \frac{4}{2} = 2$

We can next use the point slope formula to write an equation for this 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 slope we calculated and the values from the first point in the problem gives:

$\left(y - \textcolor{red}{2}\right) = \textcolor{b l u e}{2} \left(x - \textcolor{red}{3}\right)$

We can now transform this to the Standard Form for a Linear Equation. 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

$y - \textcolor{red}{2} = \left(\textcolor{b l u e}{2} \times x\right) - \left(\textcolor{b l u e}{2} \times \textcolor{red}{3}\right)$

$y - \textcolor{red}{2} = 2 x - 6$

First, we add $\textcolor{b l u e}{2}$ and subtract $\textcolor{red}{2 x}$ from each side of the equation to isolate the $x$ and $y$ variables on the left side of the equation and the constant on the right side of the equation while keeping the equation balanced:

$- \textcolor{red}{2 x} + y - \textcolor{red}{2} + \textcolor{b l u e}{2} = - \textcolor{red}{2 x} + 2 x - 6 + \textcolor{b l u e}{2}$

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

$- 2 x + y = - 4$

Now, we multiply each side of the equation by $\textcolor{red}{- 1}$ to ensure the coefficient for the $x$ variable is positive while keeping the equation balanced:

$\textcolor{red}{- 1} \left(- 2 x + y\right) = \textcolor{red}{- 1} \times - 4$

$\left(\textcolor{red}{- 1} \times - 2 x\right) + \left(\textcolor{red}{- 1} \times y\right) = 4$

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