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.

Question
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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