Question

Find the equation of the line passing through (3,-5) and whose slope is $\dfrac{7}{3}$.
a. 7x – 3y – 36 = 0
b. 7x + 3y – 36 = 0
c. 7x – 3y – 24 = 0
d. 7x – 3y + 24 = 0

Answer 1

Hint: In order to find the solution of this question, we should know a few concepts of line. An equation of line can be figured out as $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$ when we have been given $\left( {{x}_{1}},{{y}_{1}} \right)$ as one set of points through which a line is passing and m as the slope of that line.

Complete step-by-step answer:
In this question, we have been asked to find the equation of the line which is passing through (3,-5) and whose slope is $\dfrac{7}{3}$. To solve this question, we should know about a basic concept of line, that is, an equation of line is given as $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$ when we have been given $\left( {{x}_{1}},{{y}_{1}} \right)$ as one set of points through which a line is passing and m as the slope of that line.
Now, we have been given (3,-5) as the point through which the line is passing. So, we can write,
${{x}_{1}}=3$ and ${{y}_{1}}=-5$
And we have been given the slope as $\dfrac{7}{3}$. So, we can write, $m=\dfrac{7}{3}$.
Now, we will put these values in the general equation of line. So, we get,
$y-\left( -5 \right)=\dfrac{7}{3}\left( x-3 \right)$
Now, we will cross multiply the equation, so we get,
$3\left( y+5 \right)=7\left( x-3 \right)$
Now, we will open the brackets to simplify. So, we will get,
$3y+3\left( 5 \right)=7x-7\left( 3 \right)$
And we can further write it as,
$3y+15=7x-21$
Now, we will take all the terms on one side. So, we get,
$7x-3y-21-15=0$
And we can further write it as,
$7x-3y-36=0$
Therefore, we can say that the equation of the line passing through the point (3,-5) and having the slope $\dfrac{7}{3}$ is $7x-3y-36=0$. Hence, option (a) is the correct answer.

Note: We can solve this question by using the concept that, for equation of line ax + by + c = 0, if $\left( {{x}_{1}},{{y}_{1}} \right)$ is a point through which a line is passing, then $a{{x}_{1}}+b{{y}_{1}}+c=0$ and the slope of line is $-\dfrac{a}{b}$. That is, we can use the options and put the value of (3,-5) and $\dfrac{7}{3}$ as the slope to get the correct answer. But this method is very time consuming, so it is better to use the conventional method to solve this question.