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
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
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.
- 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 (?)