Question

What is the equation of a line passing through the point $\left( 4,6 \right)$ and parallel to the line $y=\dfrac{1}{4}x+4$?

Answer 1

First we will find the slope of a line $y=\dfrac{1}{4}x+4$ by comparing the equation with the general slope-intercept form of a line $y=mx+c$, where m is the slope of a line. As the parallel lines have same slope then we will use the formula of equation of a line given as $\left( y-{{y}_{1}} \right)=m\left(x-{{x}_{1}} \right)$ passing through the point $\left( {{x}_{1}},{{y}_{1}} \right)$.

Complete step by step solution:
We have been given that a line is passing through the point $\left( 4,6 \right)$ and parallel to the line $y=\dfrac{1}{4}x+4$.
We have to find the equation of a line.
Now, we know that the slope of two parallel lines will be the same as parallel lines that never intersect each other. We have an equation of a line $y=\dfrac{1}{4}x+4$.
Let us compare the given equation with the slope intercept form of a line which is given as $y=mx+c$, where m is the slope of a line. Then we will get
$\Rightarrow m=\dfrac{1}{4},c=4$
Now, we know that the equation of a line with slope m and passing through the point $\left( {{x}_{1}},{{y}_{1}} \right)$ is given by $\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right)$.
Now, substituting the values we will get
$\Rightarrow \left( y-6 \right)=\dfrac{1}{4}\left( x-4 \right)$
Now, simplifying the above obtained equation we will get
$\begin{aligned} & \Rightarrow \left( y-6 \right)=\dfrac{x}{4}-1 \\\ & \Rightarrow y=\dfrac{x}{4}-1+6 \\\ & \Rightarrow y=\dfrac{x}{4}+5 \\\ \end{aligned}$
Hence above is the required equation of a line.

Note: The point to be remembered is that parallel lines have the same slope and perpendicular lines have the product of their slopes equal to $-1$. To find the equation of a line either we need two points or one point and slope of a line.