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Hi, I'm Kendall Roberg, and today we are going to learn about direct and inverse variation.
Let's start with an example of direct variation.
Now, slices of bread and the number of sandwiches you can make are directly related if we assume that it takes two slices of bread to make a sandwich.
Let's give these two variable names. 0:00:25:000,0:00:32.000 Let's call the number of slices of bread you have "x" and the number of sandwiches you can make "y."
Let's make an x, y table.
So if I have 20 slices of bread, how many sandwiches can I make with that?
Well, each sandwich requires 2 slices of bread, so I can make 10 sandwiches before I used up all 20 slices of bread.
What if I had only 8 slices of bread?
Well, then I can make 4 sandwiches.
What if I had 100 slices of bread?
Well then I can make 50 sandwiches.
If we summarize the relationship between our variable x, slices of bread, and y, number of sandwiches, we can use the equation: the number of sandwiches we could make = half the number of slices of bread we have.
y = (1/2)x
This 1/2 we call, "the constant of variation."
x and y can vary.
The slices of bread and the number of sandwiches we can make vary, just like it did in this table.
But, the one thing that remains the same is this 1/2, the constant of variation.
In general, direct variation can be modeled with the following formula.
y = ax, where a is the constant of variation.
Now, let's look at an example of inverse variation.
Let me tell you about my friend, Johnny.
Now, Johnny is a very strong guy.
In fact, he is so strong that he can run with someone on his back.]
So someone can grab onto him and he can run.
Now he doesn't run as fast as if no one was on his back, but it's still pretty impressive that he could move at all.
Now here, we have the number of people on Johnny's back, and the speed Johnny can run in miles per hour.
If we let x be the number of people of Johnny's back, and y be the speed Johnny can run in MPH, we can model this relationship inversely...
using this formula: y = 12/x.
So, let's look at a few examples.
What if there was 1 person on Johnny's back?
Well, here we have x is 1, so 12/1 is 12.
What if there was 2 people on Johnny's back?
12/2 is 6.
What if Johnny had 4 people on his back?
12/4 = 3.
So with 4 people on his back, he could still run 3 mph.
In general, inverse variation can be modeled by this: y = a/x, where our constant of variation is a.
What's our constant of variation in our actually example?
Well, in this case, it was 12.
There is one key difference between direct and inverse variation.
With direct variation, as one variable increases, the other one does as well.
So as x increases, y increases.
Inverse variation has the opposite effect.
As one variable increases, the other variable decreases.
So, let's look at our 2 examples again.
In direct variation, as we increase the number of slices of bread, the number of sandwiches increase.
In our example with Johnny, as we increased the number of people on Johnny's back, the speed Johnny could run decreased.
Let's see if we can look at these equations and figure out if the variables are directly or inversely related.
Let's also see if we can find the constant of variation.
Now in this first example, y = 3x, we see that this is in form y = ax, so this is direct variation.
And our constant of variation is the a, in this case, 3.
In our second example here, we see that y = 4/x.
This is in the form y = a/x.
We call this inverse variation.
And our constant of variation, a, is 4.
Finally, we have y = (1/2)x.
Now actually, this example, example 1, and example 3, are in the same form: y = ax. So this is direct variation.
And our constant of variation, a, is 1/2.
Now, let's try these examples: 1, 2, and 3.
We're trying to find if it's direct or inverse variation, and what is the constant of variation.
Well, let's see...we know direct variation will be in the form y = ax, and we know inverse variation will be in the form y = a/x.
So let's see if we can put these 3 equations into one of those two forms.
Go for it.
This first one, if we divide both sides by x, we're left with y = 7/x.
We can clearly see now that y = 7/x is in the form y = a/x, so this is inverse variation.
And our constant of variation is 7.
In the second problem, let's multiply both sides by 4 to solve for y.
By doing that, we see that this is direct variation because it's in the form y = ax, where a is 4.
Now the last example, y = x + 3, is already solved for y, and we see there's really nothing we can do to get it into direct or inverse variation form.
So in this case, it's neither direct nor inverse variation.
So it's neither direct nor inverse variation...
So there's no constant of variation to talk about.
Here's an interesting example:
We're told that x and y vary inversely, and y = 7 when x = 4.
Our task is to write an equation that relates x and y.
Then, they want us to find y when x = -2.
Luckily, we have a huge hint.
We know that x and y vary inversely.
As soon as I see that word, "inversely," I know that this equation that I'm going to have to write has the form y = a/x.
This is the general form of an inverse variation equation.
So now let's use the other information that we have.
We're told that y = 7 when x = 4.
Well, let's see...
If we go back to our equation here, and we plug in a 7 for y, we know that 7 has to equal a divided by our x, 4.
So in order for this equation to be true, what would our constant of variation have to be?
Well, let's see...let's solve for a.
If I multiply both sides by 4, we end up with 28 on this side, and just a on this side.
So, we just found our constant of variation.
Our constant of variation, a, is 28.
That means our equation that relates x and y is simply y = 28/x.
This is our inverse variation equation that relates x and y.
Now let's find out what y would be when x = -2.
So we need to find y when x = -2.
Let's use our newly found equation here.
y will equal 28 divided by whatever our x is...
and it looks like our x is -2.
So 28 divided by -2 is -14.