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Question: : The equation of the line passing through the point (3, 4) and having slope 5 is \(\begin{gathere...

: The equation of the line passing through the point (3, 4) and having slope 5 is
(a)5x−y−11=0 (b)5x+y−11=0 (c)5x+y+11=0 (d)5x−y+11=0 \begin{gathered} (a)5x - y - 11 = 0 \\\ (b)5x + y - 11 = 0 \\\ (c)5x + y + 11 = 0 \\\ (d)5x - y + 11 = 0 \\\ \end{gathered}

Explanation

Solution

The equation of the line is given by the formula (y−y1)=m(x−x1)(y - {y_1}) = m(x - {x_1}),where x1,y1{x_1},{y_1} are the coordinates of the given point and m is the slope of the equation.

Complete step-by-step answer:
Step 1: It is given that the equation passes through the point (3, 4) and has a slope 5.
Let, (x1,y1)=(3,4)({x_1},{y_1}) = (3,4) and m=5
Step 2: We know that the equation of the line passing through a point (x1,y1)({x_1},{y_1}) and having a slope m.
Is (y−y1)=m(x−x1)(y - {y_1}) = m(x - {x_1})
Now let’s substitute the given point and slope in the above equation.
⇒(y−y1)=m(x−x1) ⇒(y−4)=5(x−3) ⇒(y−4)=5x−15  \Rightarrow (y - {y_1}) = m(x - {x_1}) \\\ \Rightarrow (y - 4) = 5(x - 3) \\\ \Rightarrow (y - 4) = 5x - 15 \\\
Now let’s bring the variables to one side and constants to the other’
⇒15−4=5x−y ⇒11=5x−y ⇒5x−y−11=0  \Rightarrow 15 - 4 = 5x - y \\\ \Rightarrow 11 = 5x - y \\\ \Rightarrow 5x - y - 11 = 0 \\\
Therefore the required equation is5x−y−11=05x - y - 11 = 0.
The correct option is (a)

Note: There are many methods to find the equation of the line based on the details provided
(I) if the slope, m and y intercept, c is given then the equation of the line is given by y=mx+cy = mx + c
(ii) If two points (x1,y1)and(x2,y2)({x_1},{y_1}){\text{and}}({x_2},{y_2})are given then the equation of the line joining the points is given by (y−y1)(y2−y1)=(x−x1)(x2−x1)\dfrac{{(y - {y_1})}}{{({y_2} - {y_1})}} = \dfrac{{(x - {x_1})}}{{({x_2} - {x_1})}}
(iii) When the x-intercept, a and y-intercept, b are given then the equation of the line is given by xa+yb=1\dfrac{x}{a} + \dfrac{y}{b} = 1
We must be careful when substituting the values of x1 and y1{x_1}{\text{ and }}{y_1} as there are possibilities of going wrong at that place.
We can even cross check our answer by checking the slope of the equation which we found out .We know that slope=-(coefficient of x)/(coefficient of y)
⇒m=−5−1=5\Rightarrow m = \dfrac{{ - 5}}{{ - 1}} = 5
which is the given. Hence we can use this method to cross check our answer