Writing Equations of Parallel Lines

(Videos) Writing Equations of Lines

by mathman1024

Here's a video by mathman1024 showing you how to find an equation of a line given a parallel line and a point.

$\large{\text{Summary}}$

Two lines are said to be parallel if they have the same distance between every point and they never touch. You can tell if two lines are parallel if they have the same slope.

The problem he goes over is "Find the equation of a line parallel to $y=-\frac73x+2$ and passing through $(-15,71)$." Since the line is parallel to the one given we know it will have the same slope. So already we know the slope for our line is $m=-\frac73$. Now that we have the slope and a point we can use either slope intercept formula or the point slope formula to find the equation. $$\begin{align} y-y_0&=m(x-x_0)\\ y-71&=-\frac73(x-(-15))\\ y-71&=-\frac73(x+15)\\ y-71&=-\frac73x-35\\ y-71\color{red}{+71}&=-\frac73x-35\color{red}{+71}\\ y&=-\frac73+36 \end{align}$$

Here's a video where the equation is given in standard form.

The problem he goes over is, "Find an equation of a line parallel to $8x-9y=3$ and passing though $(2,3)$." Finding the equation when given a parallel line in standard form is super easy. Just take the $Ax+By$ part and plug in the given point for $x$ and $y$. This will give you a $C$ value which will be used for the new line. $$\begin{align} 8x-9y&=\square\\ 8(2)-9(3)&=\square\\ 16-27&=-11 \end{align}$$ So the equation of the line is: $$8x-9y=-11$$

(Video) Equations of Parallel and Perpendicular Lines

by ProfRobBob

In this video, ProfRobBob goes over some problems you might see involving parallel and perpendicular lines.

$\large{\text{Summary}}$

Two lines are said to be parallel ($\parallel$) if they have the same slope.

Two lines are said to be perpendicular ($\perp$) if their slopes are negative reciprocals of each other.

Parallel Line Graphs

To find the equation of a line parallel to a different line you already know the equation for, follow the following steps:

1. Use the equation of the line you already know about to find the slope.
2. Plug the slope in for $m$ in the equation $y-y_0=m(x-x_0)$.
3. If you were given a point that needs to be on the new line, plug its $x$ and $y$ values in for $x_0$ and $y_0$. If you weren't, just choose any point and plug its values in instead.
4. Solve for $y$.

In this equation, we know that the slope our line needs to be parallel to is $2$. Thus, we know that our slope will be $2$. We also know what point the line needs to go through, so we know that our $y_0 = 2$ and $x_0 = 1$. So, we can put all this info into our equation:

$$\begin{align} y - y_0 &= m(x - x_0)\\\ \\ y - 2 &= 2(x - 1) \end{align}$$

Then, all you need to do is solve so that $y$ is alone on the left side of the equation!

$$\begin{align} y - 2 &= 2(x - 1)\\ y - 2 &= 2x - 2\\ y &= 2x - 4\\ \end{align}$$

Done!

Parallel Lines and Equations

Let's first start with the equation $y=\frac{1}{2}x+2$.

We want to make different parallel lines. When we do that, we see the slope stays the same, and the $y$-intercept changes.