Writing Equations of Parallel Lines

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.

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 $b.$ The $y$-intercept of the second line is $b=3.$ Thus, the equation is $y=2x+3.$