Question

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

Answer 1

Answer:

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

Step-by-step 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, $\\ slope = 0$.

$\\ slope = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}=0$

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, $\\ \frac{0}{0}$. 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 $\\ (-4, 6)$.

Answer 2

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