# Writing Equations of Parallel Lines

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Linear functions are a family of functions that all have a constant rate of change. There are additional ways that pairs of lines can relate. One such way is if the lines are parallel. In this section, what makes two lines parallel, and the ways in which the rules and graphs of parallel are similar, is explored.
Rule

## Parallel Lines

Lines in the same plane that never intersect are called parallel lines. Lines are parallel if and only if they have the same slope. It follows, then, that all horizontal lines are parallel to one another, as are all vertical lines. Two lines written in slope-intercept form, $y=mx+b,$ are parallel if their slopes, $m,$ are equal and they have different $y$-intercepts, $b.$

$m_1 = m_2 \quad \text{and}\quad b_1 \neq b_2$

In the diagram, it can be seen that two lines with the same slope never intersect.  Exercise

Is the line given by the equation $y=3x+5$ parallel to the line that contains the points $(1,2)$ and $(3,8)?$

Solution

For the given lines to be parallel, they must have the same slope and different $y$-intercepts. In other words, the lines must never intersect. We'll graph the lines to determine if they are parallel. Since the equation of the first line is written in slope-intercept form, we can see that $m_1=3 \quad \text{and} \quad b_1=5.$ Thus, to graph it, we will use the $y$-intercept and the slope. To draw the second line, we can plot $(1,2)$ and $(3,8)$ and connect them with a line. From the graph of the second line, we can see that it's slope is $3$ and its $y$-intercept is $(0,\text{-} 1).$ For the given lines, it has been shown that \begin{aligned} &m_1 = 3 \quad \text{and} \quad b_1 = \phantom{\text{-}} 5 \\ &m_2 = 3 \quad \text{and} \quad b_2= \text{-} 1. \end{aligned} Since $m_1=m_2$ and $b_1 \neq b_2,$ the lines are parallel.

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Exercise

Find the equation of a line which is parallel to $y=2x-5$ and passes through $(2,7).$

Solution
Two lines are parallel if they have the same slope and different $y$-intercepts. The first line, given by the equation $y=2x-5,$ is written in slope-intercept form. It can be seen that it's slope is $2$ and its $y$-intercept is $(0,\text{-} 5).$ A line parallel to this must also have the slope $m=2.$ Therefore, we can write the incomplete equation of the second line as $y=2x+b.$ This line must pass through the point $(2,7).$ We can find $b$ by substituting these coordinates for $(x,y)$ and solving for $b.$
$y=2x+b$
${\color{#009600}{7}}=2\cdot {\color{#0000FF}{2}}+b$
$7=4+b$
$3=b$
$b=3$
The $y$-intercept of the second line is $b=3.$ Thus, the equation is $y=2x+3.$
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