# A line passes through (9,3),(12,4), and (n,-5) Find the value of n.

A line passes through (9,3),(12,4), and (n,-5)
Find the value of n.
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coffentw

Given that a line passes through the points (9, 3), (12, 4), and (n, -5).
First we find the equation of the line that. passes through the points (9,3), and (12,4).
The equation of a line passing through the points $\left({x}_{1},{y}_{1}\right),\left({x}_{2},{y}_{2}\right)$ is
$\frac{y-{y}_{1}}{x-{x}_{1}}=\frac{{y}_{1}-{y}_{2}}{{x}_{1}-{x}_{2}}$
In this problem
This implies that ${x}_{1}=9,{y}_{1}=3,{x}_{2}=12,{y}_{2}=4$
. Plugging these values $\in \frac{y-{y}_{1}}{x-{x}_{1}}=\frac{{y}_{1}-{y}_{2}}{{x}_{1}-{x}_{2}}$ we get the equation of the line is
$\in \frac{y-{y}_{1}}{x-{x}_{1}}=\frac{{y}_{1}-{y}_{2}}{{x}_{1}-{x}_{2}}$
$⇒\frac{y-3}{x-9}=\frac{3-4}{9-12}$
$⇒\frac{y-3}{x-9}=\frac{1}{3}$
$⇒3\left(y-3\right)=x-9$
$⇒3y-9=x-9$
$⇒x-3y=0$
Therefore, the equation of the line is $x-3y=0$
Also given that the line passing through the point $\left(n,-5\right)$
Putting in the equation of the line we get
$x-3y=0$
$⇒n-3\left(-5\right)=0$
$⇒n+15=0$
$⇒n=-15$
Thus, the value of n is -15,