# Question #7bed9

Feb 20, 2017

$\textcolor{red}{8} x + \textcolor{b l u e}{3} y = \textcolor{g r e e n}{24}$

#### Explanation:

First, we need to find the slope of the equation. 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:

$m = \frac{\textcolor{red}{- 8} - \textcolor{b l u e}{0}}{\textcolor{red}{6} - \textcolor{b l u e}{3}} = - \frac{8}{3}$

Next, we can use the point slope formula to find 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 slope we calculate and the first point from the problem gives:

$\left(y - \textcolor{red}{0}\right) = \textcolor{b l u e}{- \frac{8}{3}} \left(x - \textcolor{red}{3}\right)$

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. We will convert the formula above to the standard form as follows:

$y = \left(\textcolor{b l u e}{- \frac{8}{3}} \times x\right) - \left(\textcolor{b l u e}{- \frac{8}{3}} \times \textcolor{red}{3}\right)$

$y = - \frac{8}{3} x + 8$

$\textcolor{red}{\frac{8}{3} x} + y = \textcolor{red}{\frac{8}{3} x} - \frac{8}{3} x + 8$

$\frac{8}{3} x + y = 0 + 8$

$\frac{8}{3} x + y = 8$

$\textcolor{red}{3} \left(\frac{8}{3} x + y\right) = \textcolor{red}{3} \times 8$

$\left(\textcolor{red}{3} \times \frac{8}{3} x\right) + \left(\textcolor{red}{3} \times y\right) = 24$

$\textcolor{red}{8} x + \textcolor{b l u e}{3} y = \textcolor{g r e e n}{24}$