Question

What is the equation of the horizontal line through (-4,6)(−4,6)left parenthesis, minus, 4, comma, 6, right parenthesis?

Answer 1

We are given two points

our point =(-4,6)

so,
`x_1=-4,y_1=6`

Since, this is horizontal line

and we know that

horizontal line will be parallel to x-axis

so, slope of horizontal must be 0

so,
`m=0`

now, we can use point slope form of line

`y-y_1=m(x-x_1)`

we can plug values

`y-6=0(x+4)`

now, we can simplify it

`y-6=0`

Add both sides by 6

`y-6+6=0+6`

`y=6`

So, equation of horizontal line is

`y=6`..............Answer

Answer 2

The equation of the horizontal line is
`\\ y = 6`.

Explanation:

The general equation for a line having the slope and some point (x, y) is represented by the formula
`\\ y - y_(1) = m(x - x_(1))`, where m is the slope (see below), and
`\\ (x_(1), y_(1))` is a given point of the line. In this case, the point given is
`\\ (x_(1), y_(1)) =(-4, 6)`.

A horizontal line has no variation in its slope (m), that is,
`\\ slope = 0`.

`\\ slope = (y_(2)-y_(1))/(x_(2)-x_(1))=0`

For this to be true, the term
`\\ y_(2) - y_(1) = 0`, or
`\\ y_(2) = y_(1)`, so
`\\ m = 0`. That is, the resulting line will be parallel to x-axis (or it could be the x-axis itself if y = 0).

A word of warning: take care that it is not the case for
`\\ x_(2) - x_(1) = 0`, in which this would result in an indeterminate form for this equation, that is,
`\\ (0)/(0)`. Or, having a variation in
`\\ y_(2) - y_(1)`≠0, with
`\\ x_(2) - x_(1)=0`, the resulting line would be a parallel line to the y-axis or the y-axis itself (if x = 0), or a line of slope = ∞.

Then, the formula:

`\\ y - y_(1)=m(x - x_(1))` could be rewritten as
`\\ y - y_(1)=0*(x - x_(1))` ⇒
`\\ y - y_(1)=0` or
`\\ y=y_(1)`, and we know that the point given is
`\\ (x_(1), y_(1))=(-4, 6)`.

So, the equation of the horizontal line through
`\\ (x_(1), y_(1)) =(-4, 6)` is then:

`\\ y = 6`.

As can be seen in the graph attached, the line is horizontal (no variation respect to y-axis
`\\ (y_(2) - y_(1))=0`, but it does for x-axis
`\\ (x_(2) - x_(1))` ≠0 ( it is different from 0 ), and the domain goes from -∞ to ∞, that is, for all values in x-axis. Notice also that the line passes through the point
`\\ (-4, 6)`.