# Question 90a6b

Oct 22, 2016

$\frac{5}{8}$

#### Explanation:

If you have two variables, and changing one results in a the other changing as well, we say that those variables are proportional. If that change is just multiplying by a constant, then we call that constant the constant of proportionality.

If we look at the equation $y = \frac{5}{8} x$, then we know $x$ and $y$ are proportional, because changing one results in the other changing as well. Furthermore, we can see that if we change $x$, then we are changing $y$ by that amount multiplied by $\frac{5}{8}$. For example, it's easy to see that if $x = 0$, then $y = \frac{5}{8} \cdot 0 = 0$. Let's see what happens to $y$ when we add $n$ to $x$:

$y = \frac{5}{8} \cdot \left(0 + n\right) = \frac{5}{8} n = 0 + \frac{5}{8} n$

So, when we increased $x$ by $n$, we increased $y$ by $\frac{5}{8} n$. We can use a similar process to show that it doesn't matter where we start. A change of $n$ in $x$ will result in a change of $\frac{5}{8} n$ in $y$. This matches our description of a constant of proportionality.

You're also right about it being the slope. In a line, the slope is just another way of expressing the change in $y$ with relation to the change in $x$. That is often the first way students are introduced to it:

"slope" = "rise"/"run" = ("change in "y)/("change in "x)#

A very fast way to find the slope of a line given the equation is to put it into point-slope form.

If a line has the equation $y = m x + b$, where $m$ and $b$ are constants, then $m$ is the slope of the line and $b$ is the $y$-intercept (that is, the line passes through the point $\left(0 , b\right)$).

Note that the given equation is actually in that form already, with $m = \frac{5}{8}$ and $b = 0$.