# How do you find an equation of a line containing the point (3, 2), and parallel to the line y - 2 = (2/3)x?

Feb 16, 2017

See the entire solution process below:

#### Explanation:

First, we must find the slope of the equation in the problem by transforming the equation to the slope-intercept form. The slope-intercept form of a linear equation is: $y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$

Where $\textcolor{red}{m}$ is the slope and $\textcolor{b l u e}{b}$ is the y-intercept value. Solving for $y$ gives:

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

$y - 0 = \left(\frac{2}{3}\right) x + 2$

$y = \textcolor{red}{\frac{2}{3}} x + \textcolor{b l u e}{2}$ Therefore:

The slope is $\textcolor{red}{m = \frac{2}{3}}$

A parallel line will have a slope the same as the line it is parallel to, for this problem $\textcolor{red}{m = \frac{2}{3}}$ We can now use the point-slope formula to find an equation for the parallel 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 determined and the point from the problem gives:

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

Or, we can solve for $y$ to put the equation in slope-intercept form:

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

$y - \textcolor{red}{2} = \frac{2}{3} x - \frac{6}{3}$

$y - \textcolor{red}{2} = \frac{2}{3} x - 2$

$y - \textcolor{red}{2} + 2 = \frac{2}{3} x - 2 + 2$

$y - 0 = \frac{2}{3} x - 0$

$y = \textcolor{red}{\frac{2}{3}} x + \textcolor{b l u e}{0}$

Or

$y = \frac{2}{3} x$

Three formulas solving this problem are:

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

$y = \textcolor{red}{\frac{2}{3}} x + \textcolor{b l u e}{0}$

$y = \frac{2}{3} x$