# How do you write an equation of a line going through (0,7), (3,5)?

Apr 24, 2017

See the entire solution process below:

#### Explanation:

First, we need to determine the slope of the line going through the two points. 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:

$m = \frac{\textcolor{red}{5} - \textcolor{b l u e}{7}}{\textcolor{red}{3} - \textcolor{b l u e}{0}} = - \frac{2}{3}$

Now, we can use the point-slope formula to find an equation going through the two points. 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 gives:

$\left(y - \textcolor{red}{7}\right) = \textcolor{b l u e}{- \frac{2}{3}} \left(x - \textcolor{red}{0}\right)$

We can also substitute the slope we calculated and the values from the second point giving:

$\left(y - \textcolor{red}{5}\right) = \textcolor{b l u e}{- \frac{2}{3}} \left(x - \textcolor{red}{3}\right)$

We can also solve the first equation for $y$ to transform the equation to 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.

$y - \textcolor{red}{7} = \textcolor{b l u e}{- \frac{2}{3}} x$

$y - \textcolor{red}{7} + 7 = - \frac{2}{3} x + 7$

$y - 0 = - \frac{2}{3} x + 7$

$y = \textcolor{red}{- \frac{2}{3}} x + \textcolor{b l u e}{7}$

Apr 24, 2017

See below.

#### Explanation:

Since a point is given at $\left(0 , 7\right)$, you know already that the y - intercept has to be 7. Therefore, you know b in the following equation:

$y = m x + b$

Now, use the slope formula to find m, the slope.

$\frac{7 - 5}{0 - 3} = \frac{2}{- 3}$

So the equation must be: $y = - \frac{2}{3} x + 7$

Hope that helps!!