Question

How do you write an equation of the line passing through $\left( { - 4, - 5} \right)$ and slope$\dfrac{{ - 3}}{2}$?

Answer 1

In this question we have to asked to find the equation, where slope and the point through which the equation passing are given, we will make use of the formula $y - {y_1} = m\left( {x - {x_1}} \right)$, where m is the slope and $\left( {{x_1},{y_1}} \right)$ is the point through which the equation passes, now substituting the given values in the formula we will get the required equation.

Complete step-by-step answer:
Given a line passing through the point $\left( { - 4, - 5} \right)$ and slope $\dfrac{{ - 3}}{2}$, and we have the equation.
So we will use the formula $y - {y_1} = m\left( {x - {x_1}} \right)$, where m is the slope and $\left( {{x_1},{y_1}} \right)$ is the point through which the equation passes,
Now here,${x_1} = - 4$,${y_1} = - 5$and$m = \dfrac{{ - 3}}{2}$,
By substituting the values in the formula we get,
$\Rightarrow y - \left( { - 5} \right) = \dfrac{{ - 3}}{2}\left( {x - \left( { - 4} \right)} \right)$,
Now simplifying we get,
$\Rightarrow y + 5 = \dfrac{{ - 3}}{2}\left( {x + 4} \right)$,
Now multiplying 2 to the both sides of the equation we get,
$\Rightarrow \left( {y + 5} \right) \times 2 = \dfrac{{ - 3}}{2}\left( {x + 4} \right) \times 2$,
Now simplifying the equation we get,
$\Rightarrow 2\left( {y + 5} \right) = - 3\left( {x + 4} \right)$,
Now multiplying for opening the brackets we get,
$\Rightarrow 2y + 10 = - 3x - 12$,
Now taking all terms to one side we get,
$\Rightarrow 2y + 10 + 3x + 12 = 0$,
Now adding the like terms we get,
$\Rightarrow 3x + 2y + 22 = 0$.
So, the required equation is $3x + 2y + 22 = 0$.

$\therefore$The equation which passes through the point $\left( { - 4, - 5} \right)$ and slope $\dfrac{{ - 3}}{2}$ is equal to $3x + 2y + 22 = 0$.

Note:
Linear equations are straight lines equations that have simple variables expressions with terms without exponents on them. There are many methods to find the equation of a line in two variables. We will use the slope-point in the questions like the given one. And other methods are slope-intercept form where a slope and y-intercept are given i.e.,$y = mx + c$, intercept form where x-intercept and y-intercept are given i.e.,$\dfrac{x}{a} + \dfrac{y}{b} = 1$, and two point’s form where two points through which line passes through them will be given, i.e.,$\dfrac{{y - {y_1}}}{{{y_2} - {y_2}}} = \dfrac{{x - {x_1}}}{{{x_2} - {x_1}}}$.