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# Graphing Horizontal and Vertical Lines

## Graphing Horizontal and Vertical Lines 1.3 - Solution

a
We want to use the slope formula to find the slope of the line that passes through the given points.

$\begin{gathered} m = \dfrac{y_2-y_1}{x_2-x_1} \end{gathered}$ In the above formula, $m$ represents the slope, and $(x_1,y_1)$ and $(x_2,y_2)$ two points on the line.

### Calculating Slope

In this exercise, we are given the points $(\text{-}3,\text{-}1)$ and $(5,\text{-}1).$ Note that, when substituting these values into the slope formula, it doesn't matter which point we choose to use as $({\color{#0000FF}{x_1}},{\color{#0000FF}{y_1}})$ or $({\color{#009600}{x_2}},{\color{#009600}{y_2}}).$ $\begin{gathered} m=\dfrac{{\color{#009600}{1}}-{\color{#0000FF}{1}}}{{\color{#009600}{5}}-({\color{#0000FF}{\text{-}3}})} \quad \text{or} \quad m=\dfrac{\phantom{\text{-}}{\color{#009600}{1}}-{\color{#0000FF}{1}}}{{\color{#009600}{\text{-}3}}-{\color{#0000FF}{5}}} \end{gathered}$ Both will give the same result. Here we will use the points in the given order and solve for the slope $m.$
$m = \dfrac{y_2-y_1}{x_2-x_1}$
$m=\dfrac{{\color{#009600}{1}}-{\color{#0000FF}{1}}}{{\color{#009600}{5}}-({\color{#0000FF}{\text{-}3}})}$
Evaluate right-hand side
$m=\dfrac{1-1}{5+3}$
$m=\dfrac{0}{8}$
$m=0$
The slope of the line that passes through the given points is $0.$ This means that as $x$ increases, $y$ neither increases nor decreases. Therefore, we have a horizontal line.

### Check by Graphing

Finally, let's plot the given points and connect them with a line to check if the direction of the slope matches what we calculated above.

Observing the graph, we can confirm that as $x$ moves in the positive direction, $y$ does not move in the positive or negative direction.

b
We'll use the slope formula to calculate the slope of the line, using the two given points.
$m=\dfrac{y_2-y_1}{x_2-x_1}$
$m=\dfrac{{\color{#009600}{0.5}}-{\color{#0000FF}{0.5}}}{{\color{#009600}{0}}-{\color{#0000FF}{2}}}$
Evaluate right-hand side
$m=\dfrac{0.5-0.5}{0-2}$
$m=\dfrac{0}{\text{-}2}$
$m=0$
A slope of $0$ means that for every $1$ horizontal step, we take $0$ vertical steps. Therefore, we have a horizontal line. The line can be graphed by drawing a line through the two points.