Graphing Vertical Lines

To graph a vertical line in the standard coordinate system, use the equation $x = k ,$ where $k$ gives the point on the $x$-axis that the line will intersect.

Suppose we want a vertical line that passes through $(5,0) .$ In addition to that point, the line will pass through

$$\ldots, (5, -3), (5, -2), (5, -1), (5,0), (5, 1), (5, 2), (5, 3), \ldots$$

and in general for any real number $Q,$ the graph will pass through $(5,Q) .$

This means the $y$-coordinate can vary to any real number, so it doesn't get fixed at all and doesn't need to appear in the vertical line equation. $x$ on the other hand must always be 5, giving an equation of $x = 5 .$

Note that if we attempt to use a more traditional format, like the slope-intercept form $y = mx + b ,$ we won't be able to form a vertical line. While the slope of $m$ could be made very large so a graphing utility makes a line look vertical, an actual vertical line has a slope that is undefined. If we think in terms of $\frac{\text{rise}}{\text{run}} ,$ the "run" of a vertical line must be zero; but this would cause division by zero, which is an impossible operation.

Note this is different from the slope of a horizontal line, which is 0. This would indicate a "rise" of 0. 0 in the numerator is fine: 0 divided by a real number (other than 0) is still 0. It's only division by zero that is prohibited.