Question

Find an equation parallel to $x = 7$ and passing through $(4, -7)$?

Answer 1

This problem deals with obtaining the equation of a line which is parallel to the line which is given by $x = 7$, here the equation whenever given that $x = a$ where here $a$ is a constant, here this equation is a straight line which is parallel to the vertical axis or the y-axis. Here on this line $x = a$, all the points on this line have the same x-coordinate which is equal to $a$, whereas the y-coordinate varies.

Complete step by step solution:
Here given that there is a point which is $(4, -7)$, and we have to find an equation of a line which passes through that point and parallel to the line $x = 7$.
Here we will be using the point-slope formula to find the equation of the line.
The point-slope formula is given by:
$\Rightarrow \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$
Here the point $\left( {{x_1},{y_1}} \right) = \left( {4, - 7} \right)$
And the slope of the line $x = 7$ is equal to $\dfrac{1}{0}$, as the line is parallel to the y-axis.
$\therefore m = \dfrac{1}{0}$
Substituting these values in the point-slope formula to get the equation of the line, as shown below:
$\Rightarrow \left( {y - \left( { - 7} \right)} \right) = \dfrac{1}{0}\left( {x - 4} \right)$
Simplifying the above equation as shown below:
$\Rightarrow \left( {y + 7} \right) = \dfrac{1}{0}\left( {x - 4} \right)$
Now cross multiply the above equation, as shown below:
$\Rightarrow 0\left( {y + 7} \right) = 1\left( {x - 4} \right)$
$\Rightarrow \left( {x - 4} \right) = 0$
Here moving the constant to the other side gives:
$\Rightarrow x = 4$
So the equation of the line is $x = 4$.

Note: Equation parallel to the y-axis or the vertical axis is always in the form of $x = a$, and the equation parallel to the x-axis or the horizontal axis is always in the form of $y = b$, which means that slope of the vertical line is equal to $\dfrac{1}{0}$, whereas the slope of the horizontal line is equal to zero. Here $a$ and $b$ may not be equal.