What is the point slope form of the line passing through: (5,7),(6,8)?

Mar 4, 2018

See a solution process below:

Explanation:

First, we need to determine the slope of the line passing 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:

$m = \frac{\textcolor{red}{8} - \textcolor{b l u e}{7}}{\textcolor{red}{6} - \textcolor{b l u e}{5}} = \frac{1}{1} = 1$

Now, we can use the point-slope formula to write the equation of 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 the first point in the problem gives:

$\left(y - \textcolor{b l u e}{7}\right) = \textcolor{red}{1} \left(x - \textcolor{b l u e}{5}\right)$

$y - \textcolor{b l u e}{7} = x - \textcolor{b l u e}{5}$

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

$\left(y - \textcolor{b l u e}{8}\right) = \textcolor{red}{1} \left(x - \textcolor{b l u e}{6}\right)$

$y - \textcolor{b l u e}{8} = x - \textcolor{b l u e}{6}$