# How do you write the standard form of a line given (7,2) and (3, -5)?

Jul 21, 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:

$m = \frac{\textcolor{red}{- 5} - \textcolor{b l u e}{2}}{\textcolor{red}{3} - \textcolor{b l u e}{7}} = \frac{- 7}{-} 4 = \frac{7}{4}$

We can now 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 $\left(\textcolor{red}{{x}_{1} , {y}_{1}}\right)$ is a point the line passes through.

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

$\left(y - \textcolor{red}{2}\right) = \textcolor{b l u e}{\frac{7}{4}} \left(x - \textcolor{red}{7}\right)$

We can now transform this equation 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

$y - \textcolor{red}{2} = \left(\textcolor{b l u e}{\frac{7}{4}} \times x\right) - \left(\textcolor{b l u e}{\frac{7}{4}} \times \textcolor{red}{7}\right)$

$y - \textcolor{red}{2} = \frac{7}{4} x - \frac{49}{4}$

$y - \textcolor{red}{2} + 2 = \frac{7}{4} x - \frac{49}{4} + 2$

$y - 0 = \frac{7}{4} x - \frac{49}{4} + \left(\frac{4}{4} \times 2\right)$

$y = \frac{7}{4} x - \frac{49}{4} + \frac{8}{4}$

$y = \frac{7}{4} x - \frac{41}{4}$

$- \textcolor{red}{\frac{7}{4} x} + y = - \textcolor{red}{\frac{7}{4} x} + \frac{7}{4} x - \frac{41}{4}$

$- \frac{7}{4} x + y = 0 - \frac{41}{4}$

$- \frac{7}{4} x + y = - \frac{41}{4}$

$\textcolor{red}{- 4} \left(- \frac{7}{4} x + y\right) = \textcolor{red}{- 4} \times - \frac{41}{4}$

$\left(\textcolor{red}{- 4} \times - \frac{7}{4} x\right) + \left(\textcolor{red}{- 4} \times y\right) = - \cancel{\textcolor{red}{4}} \times - \frac{41}{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}}}$

$\left(- \cancel{\textcolor{red}{- 4}} \times - \frac{7}{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}}} x\right) - 4 y = 41$

$\textcolor{red}{7} x - \textcolor{b l u e}{4} y = \textcolor{g r e e n}{41}$