What equation represents the line that passes through (-8, 11) and (4, 7/2)?

Jul 28, 2017

$y - 11 = - \frac{15}{24} \left(x + 8\right)$ OR $y = - \frac{5}{8} x + 6$

Explanation:

Begin by finding the slope via the formula: $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

Let $\left(- 8 , 11\right) \to \left(\textcolor{b l u e}{{x}_{1}} , \textcolor{red}{{y}_{1}}\right)$ and $\left(4 , \frac{7}{2}\right) \to \left(\textcolor{b l u e}{{x}_{2}} , \textcolor{red}{{y}_{2}}\right)$ so,

$m = \frac{\textcolor{red}{\frac{7}{2} - 11}}{\textcolor{b l u e}{4 - \left(- 8\right)}}$

$m = \frac{\textcolor{red}{\frac{7}{2} - \frac{22}{2}}}{\textcolor{b l u e}{4 + 8}} \leftarrow$ Find LCD for $\frac{7}{2}$ and $11$ and simplify

$m = \frac{\textcolor{red}{- \frac{15}{2}}}{\textcolor{b l u e}{12}} = - \frac{15}{2} \cdot \frac{1}{12} \leftarrow$ Apply the rule: $\frac{\frac{a}{b}}{c} = \frac{a}{b} \cdot \frac{1}{c}$ and multiply

$m = - \frac{15}{24}$

Now that we have found the slope, we can find the equation of the line using the point-slope formula: $y - {y}_{1} = m \left(x - {x}_{1}\right)$

Where $m$ is the slope (which we just found) and ${x}_{1}$ and ${y}_{1}$ are the $x$ and $y$ values of either of the two given points. Substituting this information, we can easily find the equation of the line.

Recall that the slope, or $m$, Is $- \frac{15}{24}$ and ${x}_{1}$ and ${y}_{1}$ are the $x$ and $y$ values of either of the two given points. I will choose to use the point $\left(- 8 , 11\right)$ as my ${x}_{1}$ and ${y}_{1}$ values just because I don't want to deal with the fraction. Just know that the point $\left(4 , \frac{7}{2}\right)$ will work just as well.

The equation of the line:

$y - \left(11\right) = - \frac{15}{24} \left(x - \left(- 8\right)\right)$

$y - 11 = - \frac{15}{24} \left(x + 8\right)$

Note: We could leave the equation above as is and say that this is the equation of the line. We could also express the equation in $y = m x + b$ form if desired in which case we must solve the equation for $y$

Solving for $y$ would give us: $y = - \frac{5}{8} x + 6$

Below is what the line looks like along with the two points given in the problem.