Question

Find an equation of a line that passes through the point (3, - 4) and parallel to the y-axis.

Answer 1

Hint: When the line is parallel to the y-axis the slope of the line is undefined. Using the formula of an equation of line having a slope and passing through a point \[y - {y_1} = m(x - {x_1})\] we get the required equation.

Complete step-by-step solution:
We are given a point $(3, - 4)$
We have to find an equation of a which is parallel to the y-axis and passing through a point $(3, - 4)$.
As we know that when the line is parallel to the y-axis the x-coordinate of the line remains the same, it will not vary throughout the line.
If we look at the formula of the slope of two points, $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$, the values of $x$ come in the denominator and our line have the same value of $x$ throughout the line it means the denominator will be $0$.

Hence, the slope of the required line is undefined as the denominator is $0$.
Assume the slope with the denominator $0$ and any number in the numerator let say $1$ because it does not affect the slope of the required line.
Therefore, \[m = \dfrac{1}{0}\]
Now use the formula of an equation of a line.
We know that equation of a line passing through a point $({x_1},{y_1})$ and slope $m$ is:
\[y - {y_1} = m(x - {x_1})\]
Substitute the value of \[m = \dfrac{1}{0}\] and $({x_1},{y_1}) = (3, - 4)$in the formula to get the equation of line.
$y + 4 = \dfrac{1}{0}(x - 3)$
Cross multiply and solve the equation.
$
  0(y + 4) = x + 3 \\
  x + 3 = 0 \\
 $
As we can observe that there is no component of y in our required equation of a line. It means it is parallel to the y-axis.
Hence, the equation of a line is $x = 3$
Note: The line which is parallel to the y-axis is a vertical line and its slope is undefined.
We can directly write the equation of the vertical line by $x = a$ where $a$ is the point lies on the line.