What Is the Difference Between a Direct and an Inverse Relationship? | Sciencing
Inverse Relationships are written as A = k. B where k is a nonzero constant. Direct Relationship Examples: Radius of a circle and its area are directly related. Two values are said to be in direct proportion when increase in one results in an increase in the other. Similarly, they are said to be in indirect proportion when. Direct variation describes a simple relationship between two variables. We say y varies directly with x (or as x, in some textbooks) if.
So sometimes the direct variation isn't quite in your face. But if you do this, what I did right here with any of these, you will get the exact same result. Or you could just try to manipulate it back to this form over here. And there's other ways we could do it. We could divide both sides of this equation by negative 3. And now, this is kind of an interesting case here because here, this is x varies directly with y.
Or we could say x is equal to some k times y. And in general, that's true. If y varies directly with x, then we can also say that x varies directly with y. It's not going to be the same constant. It's going to be essentially the inverse of that constant, but they're still directly varying.
Now with that said, so much said, about direct variation, let's explore inverse variation a little bit.
Inverse variation-- the general form, if we use the same variables. And it always doesn't have to be y and x. It could be an a and a b. It could be a m and an n. If I said m varies directly with n, we would say m is equal to some constant times n. Now let's do inverse variation. So let me draw you a bunch of examples.
And let's explore this, the inverse variation, the same way that we explored the direct variation. And let me do that same table over here. So I have my table. I have my x values and my y values. If x is 2, then 2 divided by 2 is 1. So if you multiply x by 2, if you scale it up by a factor of 2, what happens to y? You're dividing by 2 now. Here, however we scaled x, we scaled up y by the same amount.
Now, if we scale up x by a factor, when we have inverse variation, we're scaling down y by that same. So that's where the inverse is coming from. And we could go the other way.
So if we were to scale down x, we're going to see that it's going to scale up y. So here we are scaling up y. So they're going to do the opposite things.
And you could try it with the negative version of it, as well. So here we're multiplying by 2. And once again, it's not always neatly written for you like this. It can be rearranged in a bunch of different ways.
But it will still be inverse variation as long as they're algebraically equivalent. So you can multiply both sides of this equation right here by x. And you would get xy is equal to 2.
- Direct and inverse proportions
This is also inverse variation. You would get this exact same table over here.
Intro to direct & inverse variation
You could divide both sides of this equation by y. So notice, y varies inversely with x. And you could just manipulate this algebraically to show that x varies inversely with y.
So y varies inversely with x. This is the same thing as saying-- and we just showed it over here with a particular example-- that x varies inversely with y. And there's other things.
Direct and inverse proportions | Class 8 (India) | Math | Khan Academy
We could take this and divide both sides by 2. There's all sorts of crazy things. And so in general, if you see an expression that relates to variables, and they say, do they vary inversely or directly or maybe neither? You could either try to do a table like this. If you scale up x by a certain amount and y gets scaled up by the same amount, then it's direct variation. If you scale up x by some-- and you might want to try a couple different times-- and you scale down y, you do the opposite with y, then it's probably inverse variation.
Whether you have a specific scientific question in mind such as: What happens to the global temperature if the amount of carbon dioxide in the atmosphere increases, or how does the strength of gravity vary when you move further away from the source, or you are more interested in an abstract mathematical setting, finding out the difference between direct and inverse relationships is essential if you want to describe these relationships.
In short, direct relationships increase or decrease together, but inverse relationships move in opposite directions. A bigger diameter means a bigger circumference.
Direct and inverse relationships - Math Central
In an inverse relationship, an increase in one quantity leads to a corresponding decrease in the other. Faster travel means a shorter journey time.
How Does y Vary with x? Scientists and mathematicians dealing with direct and inverse relationships are answering the general question, how does y vary with x? Here, x and y stand in for two variables that could be basically anything. By convention, x is the independent variable and y is the dependent variable. So the value of y depends on the value of x, not the other way around, and the mathematician has some control over x for example, she can choose the height from which to drop the ball.
When there is a direct or inverse relationship, x and y are proportional to each other in some way. Direct Relationships A direct relationship is proportional in the sense that when one variable increases, so does the other. Using the example from the last section, the higher from which you drop a ball, the higher it bounces back up. A circle with a bigger diameter will have a bigger circumference. If you increase the independent variable x, such as the diameter of the circle or the height of the ball dropthe dependent variable increases too and vice-versa.