# How do you write an equation in standard form given a line that passes through (5,8) and (2,2)?

Mar 4, 2018

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:

$m = \frac{\textcolor{red}{2} - \textcolor{b l u e}{8}}{\textcolor{red}{2} - \textcolor{b l u e}{5}} = \frac{- 6}{-} 3 = 2$

We can now use the point slope formula to write an equation for the line. The point-slope form of a linear equation is: $\left(y - \textcolor{b l u e}{{y}_{1}}\right) = \textcolor{red}{m} \left(x - \textcolor{b l u e}{{x}_{1}}\right)$

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

Substituting the slope we calculated and the values from either point in the problem (I will use the values from the second point) into the formula gives:

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

We can now solve this equation for the Standard Form of 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

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

$y - \textcolor{b l u e}{2} = 2 x - 4$

$- \textcolor{red}{2 x} + y - \textcolor{b l u e}{2} + 2 = - \textcolor{red}{2 x} + 2 x - 4 + 2$

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

$- 2 x + y = - 2$

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

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

$2 x + \left(- 1 y\right) = 2$

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