# How do you write an equation of a line that passes through points (0,5), (-3,5)?

##### 2 Answers
Feb 9, 2017

$y = 5$

#### Explanation:

The equation of a line can be written as

$y - {y}_{0} = m \cdot \left(x - {x}_{0}\right)$

where $\left({x}_{0} , {y}_{0}\right)$ is any point that lies on the line.

The gradient, $m$, of the line can be found using any two non-identical points that lie on the line.

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

$= \frac{5 - 5}{- 3 - 0}$

$= 0$

A gradient of zero indicates that the line is horizontal.

The equation of the line can thus be simplified to

$y = {y}_{0}$

In this case, both points have a $y$ coordinate of 5. The equation of this line is therefore

$y = 5$.

Feb 9, 2017

$\textcolor{g r e e n}{y = 5}$

#### Explanation:

For this particular example, we could note that the value of the $y$ coordinate is a constant: $5$, so the equation is:
$\textcolor{w h i t e}{\text{XXX}} y = 5$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For the general case:
$\textcolor{w h i t e}{\text{XXX}}$Given two points color(red)(""(x_1,y_1) and color(blue)(""(x_2,y_2))
$\textcolor{w h i t e}{\text{XXX}}$A two-point equation can be written as:
$\textcolor{w h i t e}{\text{XXXXXX}} \frac{y - \textcolor{red}{{y}_{1}}}{x - \textcolor{red}{{x}_{1}}} = \frac{\textcolor{red}{{y}_{1}} - \textcolor{b l u e}{{y}_{2}}}{\textcolor{red}{{x}_{1}} - \textcolor{b l u e}{{x}_{2}}}$

Substituting
$\textcolor{w h i t e}{\text{XXX")color(red)(} \left(0 , 5\right)}$ for color(red)(""(x_1,y_1)) and
$\textcolor{w h i t e}{\text{XXX")color(blue)(} \left(- 3 , 5\right)}$ for color(blue)(""(x_2,y_2))
we have:
$\textcolor{w h i t e}{\text{XXX")(y-color(red)5)/(x-color(red)0)=(color(red)5-color(blue)(5))/(color(red)(0)-color(blue)(} \left(- 3\right)}$
Simplifying:
$\textcolor{w h i t e}{\text{XXX}} \frac{y - 5}{x} = 0$

$\textcolor{w h i t e}{\text{XXX}} y = 5$