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Question: Find the equation of the line passing through (3,-5) and whose slope is \(\dfrac{7}{3}\). a. 7x – ...

Find the equation of the line passing through (3,-5) and whose slope is 73\dfrac{7}{3}.
a. 7x – 3y – 36 = 0
b. 7x + 3y – 36 = 0
c. 7x – 3y – 24 = 0
d. 7x – 3y + 24 = 0

Explanation

Solution

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−y1=m(x−x1)y-{{y}_{1}}=m\left( x-{{x}_{1}} \right) when we have been given (x1,y1)\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 73\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−y1=m(x−x1)y-{{y}_{1}}=m\left( x-{{x}_{1}} \right) when we have been given (x1,y1)\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,
x1=3{{x}_{1}}=3 and y1=−5{{y}_{1}}=-5
And we have been given the slope as 73\dfrac{7}{3}. So, we can write, m=73m=\dfrac{7}{3}.
Now, we will put these values in the general equation of line. So, we get,
y−(−5)=73(x−3)y-\left( -5 \right)=\dfrac{7}{3}\left( x-3 \right)
Now, we will cross multiply the equation, so we get,
3(y+5)=7(x−3)3\left( y+5 \right)=7\left( x-3 \right)
Now, we will open the brackets to simplify. So, we will get,
3y+3(5)=7x−7(3)3y+3\left( 5 \right)=7x-7\left( 3 \right)
And we can further write it as,
3y+15=7x−213y+15=7x-21
Now, we will take all the terms on one side. So, we get,
7x−3y−21−15=07x-3y-21-15=0
And we can further write it as,
7x−3y−36=07x-3y-36=0
Therefore, we can say that the equation of the line passing through the point (3,-5) and having the slope 73\dfrac{7}{3} is 7x−3y−36=07x-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 (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) is a point through which a line is passing, then ax1+by1+c=0a{{x}_{1}}+b{{y}_{1}}+c=0 and the slope of line is −ab-\dfrac{a}{b}. That is, we can use the options and put the value of (3,-5) and 73\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.