# How to find slope intercept through parallel lines?

I'm working on a school project. I need the slope intercepts from each problem. The problems are:

Through: (5, 1) parallel to y=1/5x-1

Through: (-1,-3) parallel to y=2x+4

And so a few others.

I don't want them solved for me, I want a text only tutorial on how to solve it myself. I'd much rather learn it instead of having it handed to me.

• parallel lines have the same slope, you know that the line equation is $y=mx+c$, now plug in values for $y,x,m$ and find $c$ Nov 7, 2017 at 19:02

Let's do it by example. Say we want the slope-intercept form of the line through $(1, 7)$ and parallel to $y = 3x - 6$. Slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept; so we just want to find the slope and the $y$-intercept of this new line.

The word "parallel" means "having the same slope", when we're talking about lines. So the slope of our line has to be the same as the slope of $y = 3x - 6$. Conveniently, $y = 3x - 6$ is already written in slope-intercept form - so we can just read off the slope, which is $3$. So our line should be $y = 3x + b$, where $b$ is the $y$-intercept.

Now, we could hunt down the $y$-intercept - or we could just make use of the fact that $(1,7)$ is on our line. To say "$(a,b)$ is on the line" is the same as saying "when you plug in $a$ for $x$, $y$ is $b$". So when we plug in $1$ for $x$, $y$ must be $7$. So, $7 = 3 \cdot 1 + b$. Solving for $b$ gives us $4$, so our answer is $4$.

I should say, there's another option - we could also make use of point-slope form. If you're not familiar with point-slope form, stop reading, because the next paragraph won't be helpful to you.

Point-slope form is $y - y_0 = m(x - x_0)$, where $(x_0,y_0)$ is a point on the line and $m$ is the slope. In this case, we knew that the slope was $3$ and $(1,7)$ was on the line, so the point-slope form of the line was $y - 7 = 3(x - 1)$. Distributing the $3$, we have $y - 7 = 3x - 3$. Adding $7$ to both sides, we have $y = 3x + 4$. This is slope-intercept form, so we're done (again).

HINTS

1. Parallel lines have the same slope but different intercepts. Find the slope first.
2. A line that passes through $(a,b)$ must have the equation $$y-b = m(x-a)$$ with slope $m$ and variables $x,y$.