Problem: The equation of the vertical line that passes through the point (4, -3).

Solution: **x = 4**** **

To understand **why**, please read the following step by step solution

1. **Read**, **understand**
the situation, **identify** and pull out **important information.**

- There are infinite lines passing through the point (4,-3).

- There is
**only one line vertical **to **y-axis**
passing through the point (4,-3). This line is **horizontal**,** **parallel to x-axis, and with all its points having the same "y"* (-3)*. This line crosses the
y-axis in the y-intercept point.

- There is
**only one line vertical **to** x-axis** passing through the point (4,-3),
*parallel* to y-axis, with all its points having the same "x"
*(4)*. It crosses the x-axis in the x-intercept point.

2. **Translate** each of the **keywords**
in the problem to their appropriate **mathematical ****symbols**.

Every point of a line is identified by its coordinates (x,y). All points of this vertical line passing through (4,-3) have the same x coordinate “4”. **x = 4.**

3. **Set up** and **solve**
the equation:

**x = 4** The equation of the vertical line passing through (4,-3). It's true only when the value of coordinate x is “4”,
**for any value of "y"**.

4. **Verify** your answer.

- The
**slope (m) of a vertical line is undefined .** Vertical lines have
no slope (it does not exist in definite concepts)!

**m = RISE /RUN;**

RISE = any;

RUN = 0 because when going from point (4,-3) to another point on the vertical line *x = 0*, as we don't move horizontally (just vertically)!

slope (m) = RISE/RUN; **m = RISE / 0** ; **the slope is
****undefined**

*So our line represented by ***x =**** 4** is a vertical line!

- The equation of a vertical line is a special case of the standard form equation where

A = 0, B = 1, C = 4

0y + x = 4;

**x = 4** Represents the standard form equation of the requested vertical line. No matter what the y-value is, the x-value is always a constant value “4”, “x” does not change: varying "y" we move vertical to x-axis.

5. **Curiosities**:

- Vertical lines can’t be written in slope-intercept form (the slope is undefined and there is no y-intercept).

- The
**x-intercept point** of our vertical line is (4,0).

**Graphing** of our vertical line x = 4: Plot the given point (4,-3) and the x-intercept point (4,0), and draw a line through the points (4,-3) and (4,0).

- In 2-dimension geometry, vertical lines have not y-intercept.

**Mathematically**, the only **vertical line having infinite y-intercept points** is
**x = 0**, because its points coincide with the points of y-axis. The graph of the line is the y-axis, and every real number
*for y* could be considered as a y-intercept.

- In a
**3-dimensional geometry** a vertical line has
**only one y-intercept**! It does not need be perpendicular to the x-axis nor parallel to the y-axis.

- Mathematically, two parallel lines intersect at the infinity! Find the y-intercept point. Think for a while (?)