# Proportional and Non-Proportional Linear Relationships

## Proportional and Nonproportional Linear Relationships

In math, we deal with proportional and non-proportional linear relationships. A proportional relationship has the general structure of Y equals KX. Y and X are variables. K is a constant. Say we’re trying to figure out how far someone drove. Y here is going to equal distance. K, our constant is going to equal 30 miles per hour. That’s how fast they were driving. Our other variable, x, is going to equal the amount of hours driven.

We’re trying to find out how far they drove. We plug in our constant, miles over hours, and then we multiply that by 2 hours. Hours and hours cross out and we get 60 miles. They drove 60 miles. Now, we have two variables here. Right here, this variable was 2. Now, say we double it. Say we now have 4 right there. 30 times 4 is going to be 120 miles. Notice here that the X doubled, but then Y also doubled.

This is a proportional relationship, because if X increases by a factor of 2 then Y increases by a factor of 2. If X decreases by a factor of 7, then Y is going to decrease by a factor of 7. These are directly proportional to each other, because K is always the same. That’s a proportional relationship. I’m going to get some of this out of the way and we’re going to take a look at a non-proportional relationship, which has the general setup of Y equals AX plus B.

This time we have two constants. We have A and B as our constants. We have Y and X as our variables. Say we’re trying to figure out a total phone bill. Y here is going to stand for our monthly bill. A is going to stand for our permanent charge of 25 cents per minute. X is going to stand for the number of minutes. Then we’re going to add that to just the overall monthly charge of 10 dollars. In this case, we’re not going to use the money symbol. We have the set up here.

We have two constants. This 0.25 per minute charge is always going to be there and the monthly charge is going to be there of 10 dollars. The things that are going to change are X, the amount of minutes you use, and then because the amount of minutes is going to change, the overall cost is going to change. This is a non-proportional relationship. If say, for example, X equals 4. 4 times 0.25 equals 1 plus 10 equals 11. Y equals 11. Say this equals 8.

This number is doubled, so now it’s going to equal 2, plus 10 equals 12. Y equals 12. Notice here what happened. X doubled, X increased by a factor of 2, but Y just increased by 1. It didn’t increase by 2. It didn’t even come close. It just increased by one number. That’s a non-proportional relationship. The reason it’s non-proportional is because of this extra constant. That’s what messes it up. Right here we have four different equations.

We want to identify whether these are proportional or non-proportional. This one right here, we see we have two variables and we have a constant, so this must mean it’s proportional. We come to this one, we have Y equals -2x plus 1.

Notice we have two variables, but then we have two constants. We have -2 here plus 1, so this must mean non-proportional. Here we have two variables and one constant, so it’s proportional. Notice the difference here between proportional and non-proportional is the number of constants. When you add that second constant, then you begin to have a non-proportional relationship as we see here with the two constants. That’s the difference between a proportional and a non-proportional linear relationship.