# Is this a linear relationship and is this equation valid?

*This is not a homework question

Let's say you are given a pattern for a bouquet of flowers (I write it out since I can't actually draw it):

-Boquet 1: 2 red flowers, 1 white flower
-Boquet 2: 4 red flowers, 2 white flower
-Boquet 3: 6 red flowers, 3 white flower


I know you can create a linear relationship of the number of white flowers for a bouquet number. Like, $Total Flowers=3 \cdot Bouquet Number$. Or for individual flowers, i.e $RedFlowers = 2 \cdot bouquet Number$. Of course you can graph these linear relationships.

But is it possible to create a relationship for the number of red flowers, to white flowers and create an equation for that, and then graph that as a linear relationship as well? For example is this equation valid, $White Flower={RedFlower \over 2}$? And then graph where white flower is the y axis, and red flower is the x axis?

Which basically creates:

-2 red Flowers, 1 white flower
-4 red Flowers, 2 white flower
-6 red Flowers, 3 white flower


What is making making me say yes is is the fact that there is a clear relationship between the number of red flower to white flowers, and we can see that for a given bouquet, the number of white flowers equals half of the number of red flowers in a given bouquet. However, what is making me say no (which I don't know if it is right or not), is that in the above equation, white flowers is a function of red flowers, instead of flowers as being a function of the bouquet numbers; so, you are taking away the bouquet number, as a result there is no pattern, and therefore you can't create an equation with it (as the order is not specified if there is no bouquet number for the independent variable x)? Instead what you are creating is a scatterplot of the ratios of red flowers, to white flowers when you put the points on a graph. So the above equation isn't valid.

What's also making me say no is that white flowers will never equal red flowers, and so it is not an equation. You can do this with bouquet numbers, because they contain the flowers, but you can't directly create an equation with white flowers being half of red flowers, since red flowers and white flowers are different? Since there is no equation (as white flowers is not a function of red flowers), you can't say there is a linear relationship, and so you can't graph that equation. Instead what you are creating is just ratios?

Also, aren't you just plotting the ratio of red flowers to white flowers? Since you are just plotting ratios, there is no linear relationship?

Can someone help me? I'm so confused.

• Don't be confued just because the "cause" bouquet number gets eliminated! Functions are only cum hoc, not necessarily propter hoc, so to speak. – Hagen von Eitzen Feb 5 '14 at 21:37

## 1 Answer

There are lots of linear relationships that you have here:

Number of red flowers $R$ in terms of number of white flowers $W$: $R = 2W$

Number of white flowers in terms of number of red flowers: $W = R/2$

Total number of flowers $T$ in terms of the number of red and white flowers: $T = R + W$

Total number of flowers $T$ in terms of the number of red flowers: $T = 3R/2$

Total number of flowers $T$ in terms of the number of white flowers: $T = 3W$

Number of white flowers in terms of the total number of flowers: $W = T/3$

Number of red flowers in terms of the total number of flowers: $R = 2T/3$

They all work, and they're all linear (in one or more variables).