Question

How do you write the slope-intercept form of the equation of the line perpendicular to the graph of $y=-\dfrac{3}{2}x-7$ that passes through (3,-2)?

Answer 1

This type of problem is based on the concept of lines. First, we have to consider the slope-intercept form of the equation of the line, that is, $y=mx+c$. Here the line is perpendicular to $y=-\dfrac{3}{2}x-7$. Then we need to find the slope of $y=-\dfrac{3}{2}x-7$. We should then take the negative reciprocal of the given equation and substitute in m. Thus, we obtain the slope intercept form of the equation of the line with necessary calculations.

Complete step-by-step answer:
According to the question, we are asked to find the equation of the line in slope-intercept form.
We have been given the line $y=-\dfrac{3}{2}x-7$. --------(1)
From equation (1), the slope of the equation is
$m=-\dfrac{3}{2}$.
We first have to find the slope-intercept form, that is, $y=mx+c$. ------(2)
But we have been given that the line $y=-\dfrac{3}{2}x-7$ is perpendicular to $y=mx+c$.
We know that the product of slopes of two perpendicular lines is equal to -1.
$-\dfrac{3}{2}m=-1$.
$\Rightarrow -\dfrac{3m}{2}=-1$
$\Rightarrow \dfrac{3m}{2}=1$
On further simplification, we get,
$\dfrac{3m}{2}\times 2=1\times 2$
$\Rightarrow 3m=2$
$\Rightarrow \dfrac{3m}{3}=\dfrac{2}{3}$
$\therefore m=\dfrac{2}{3}$
Now, we have to substitute m in equation (2).
$\Rightarrow y=\dfrac{2}{3}x+c$ --------(3)
But we have been given that the line passes through (3,-2).
Therefore,
$3=\dfrac{2}{3}\left( -2 \right)+c$
$\Rightarrow \dfrac{-4}{3}+c=3$
$c=3+\dfrac{3}{4}$
On further simplification, we get,
$c=\dfrac{3\times 4}{4}+\dfrac{3}{4}$
$\Rightarrow c=\dfrac{12}{4}+\dfrac{3}{4}$
$\Rightarrow c=\dfrac{12+3}{4}$
$\Rightarrow c=\dfrac{16}{4}$
$\therefore c=4$

Now, substitute c in equation (3).
We get,
$y=\dfrac{2}{3}x+4$
Therefore, the slope-intercept form is $y=\dfrac{2}{3}x+4$..

Hence, the slope-intercept form of the equation of the line perpendicular to the graph of $y=-\dfrac{3}{2}x-7$ that passes through (3,-2) is $y=\dfrac{2}{3}x+4$.

Note: Whenever you get this type of problem, we should make sure about the formula of slope-intercept equation. We should avoid calculation mistakes based on sign conventions. We should be thorough with the formulas of all forms of line equations. It is always advisable to find the slope with the given equation first and then solve the rest of the part. We should not get confused with the product of slopes in parallel and perpendicular planes.
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