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## The equation of the horizontal line that passes through the point (-2, -1).

The answer I got is  x= -2. Am i right?

NO! The answer is y = -1. To understabd why, please read the following STEP BY STEP SOLUTION

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

• There are infinite lines passing through the point (-2,-1).
• There is only one horizontal line parallel to y-axis passing through the point (-2,-1). This line is vertical to x-axis, and crossing the x-axis in the x-intercept point.
• There is only one horizontal line parallel to x-axis passing through the point (-2,-1). This line is vertical to y-axis, and crossing the y-axis in the y-intercept point. This is our requested parallel line passing through the point (-2,-1).
• A line is a set of infinite points! Every point of a line is identified by its coordinates (x,y). All points of this horizontal line passing through (-2,-1) have the same “y”- coordinate .

2. Translate each of the keywords in the problem to their mathematical symbols.
“Horizontal line” through (-2,-1): all points have same y = -1.

3. Set up and solve the equation:

y = -1    It's the equation of the horizontal line passing through (-2,-1). This equation is true only when the value of the coordinate y  is “-1”, for any value of "x".

• The slope (m) of an horizontal line is 0!
m = RISE /RUN;     m = 0 / RUN = 0
RUN = Any
RISE = 0;   as you move to the right along the line, it does not rise or fall at all. In fact, when going from point P1 (-2,-1) to another point P2 (x2, y2) of the given horizontal line:

RISE = y2 – y1

RISE = (-1) - (-1) =

RISE = -1 + 1

RISE = 0.

Therefore, m = 0 / RUN

m = 0

• The equation of an horizontal line is a special case of slope-intercept form having m = 0, and the y-intercept = b

y = mx + b;

y = 0x + (-1);

y = -1    (No matter what the x-value is, the y-value is always a constant value “-1”;   “y” does not change).

• In the point-intercept form y - y1 = m (x - x1) we get the same result:

y - (-1) = 0 (x -  (- 2));

y + 1 = 0 ;

y = -1

5. Curiosities:

Standard form equation Ax + By = C, for the horizontal line through (-2,-1):

0x + 1y = -1. Where A=0, B=1, C=-1 (in our case).

• The y-intercept point of our horizontal line is (0,-1).

Graphing our horizontal line y = -1: Plot the given point (-2,-1) and the y-intercept point (0,-1). Draw a line through the points (-2,-1) and (0,-1).

• In 2-dimension geometry, horizontal lines have not x-intercept in the conventional sense of the words (Euclidean plane) .

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

• According to the Euclid’s parallel postulate, two parallel lines do not intersect! However mathematically, in a non-Euclidean space, parallel lines intersect only at the infinity!  So, find the x-intercept point for our given horizontal line. Think for a while (?)

A horizontal line runs left to right or vice versa like so

<--------------------------->

which means that the line will cross on the y axis and the point -1 lies on the y axis. This line will never touch the x axis. It will run straight 1 unit below the x axis forever and crossing only y.

No, and this is why.

If you were to plot that point, you would start at the origin and go left two, down one.

Looking at the Cartesian Plane, a horizontal line going through that point would occur at  y = -1