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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?

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3 Answers

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

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".

4. Verify your answer.

• 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.