Question

How do you find an equation of the line having the given slope $m=\dfrac{6}{7}$ and containing the given point $\left( 6,-6 \right)$ ?

Answer 1

We are given the slope and a point of a straight line and we have to find the equation of that straight line. Therefore, we shall form the equation using the slope point form of the equation of a straight line. Then, we will modify the equation further to bring it to its simplest form.

Complete step-by-step solution:
Since we are given the slope and one point of the straight line, therefore, we will use the slope-point of the equation of a straight line.
$y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$
Where,
$m=$ slope of the straight line
$\left( {{x}_{1}},{{y}_{1}} \right)=$ point on the straight line
Here, we have slope, $m=\dfrac{6}{7}$ and point $\left( {{x}_{1}},{{y}_{1}} \right)=\left( 6,-6 \right)$.
Substituting these values, we get
$y-\left( -6 \right)=\dfrac{6}{7}\left( x-6 \right)$
$\Rightarrow y+6=\dfrac{6}{7}\left( x-6 \right)$
We will now multiply 7 on both sides,
$\Rightarrow 7\left( y+6 \right)=7\times \dfrac{6}{7}\left( x-6 \right)$
$\Rightarrow 7\left( y+6 \right)=6\left( x-6 \right)$
Opening the brackets and multiplying terms on left hand side with 7 and on right hand side with 6, we get
$\Rightarrow 7y+42=6x-36$
We shall now transpose 42 to the right-hand side and add with the other constant term.
$\begin{aligned} & \Rightarrow 7y=6x-36-42 \\\ & \Rightarrow 7y=6x-78 \\\ \end{aligned}$
Further, transposing $7y$ to the right-hand side and rearranging the equation, we get
$\Rightarrow 0=6x-7y-78$
$\Rightarrow 6x-7y-78=0$
Therefore, the equation of the line having the given slope $m=\dfrac{6}{7}$ and containing the given point $\left( 6,-6 \right)$ is given as $6x-7y-78=0$.

Note: One must have prior knowledge of all the forms of equations of a straight line. The most common forms of the equation of straight line are slope-intercept form, slope-point form, two-point form, intercept form and parametric form. However, the standard form of a linear equation in two variables is given as $ax+by+c=0$ where, $a$, $b$and $c$ are constants.