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# 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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Vectors and spaces asked 2020-10-26
A line passes through (9,3),(12,4), and (n,-5)
Find the value of n.

## Answers (1) 2020-10-27
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 $$(x_{1}, y_{1}), (x_{2}, y_{2})$$ is
$$\frac{y-y_{1}}{x-x_{1}}=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}$$
In this problem $$(x_{1}, y_{1}) = (9, 3),\ and\ (x_{2}, y_{2}) = (12,4).$$
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
$$\in \frac{y-y_{1}}{x-x_{1}}=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}$$
$$\Rightarrow \frac{y-3}{x-9}=\frac{3-4}{9-12}$$
$$\Rightarrow \frac{y-3}{x-9}=\frac{1}{3}$$
$$\Rightarrow 3(y - 3) = x - 9$$
$$\Rightarrow 3y - 9 = x - 9$$
$$\Rightarrow x - 3y = 0$$
Therefore, the equation of the line is $$x - 3y = 0$$
Also given that the line passing through the point $$(n, -5)$$
Putting $$x = n,\ and\ y = -5$$ in the equation of the line we get
$$x - 3y = 0$$
$$\Rightarrown - 3(-5) = 0$$
$$\Rightarrow n + 15 = 0$$
$$\Rightarrow n = -15$$
Thus, the value of n is -15,

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