Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. When referring to variation, we're referring to the way variables interact with direct variation. Y is going to equal some constant of variation times the variable X. That constant of variation once we have a problem will stay constant throughout the problem. Sometimes people think of Y equal K * X as being a linear equation with AY intercept of 00. Inverse variation Y is going to equal that constant of variation again times 1 / X, or the inverse of the variable that would equal K / X joint variation. Every problem can be different and it's a variable that varies as the product of a product or quotient of other variables. So Y is going to equal K * a product in quotient of other variables. Let's look at a few examples. If Y varies directly as XY equal 10 when X = 8. So the first thing we're going to do if Y varies directly, we know Y equal K * X. Well now Y is 10 when X is 8 and we can actually find the constant. So the K is going to be 10 / 8 or the K reduces to 5 fourths. So we can now have an equation where we have a known constant of variation Y equal 5 fourths times X. The constant never changes. So the second part of this says find Y when X equal 2. So now we're just going to put 2IN for our X and solve for Y. So Y is five halves when X is 2. Let's look at the same problem but the change it to inverse this time. So Y equal K / X so 10 equal K / 8. We find our constant of variation in this case to be 80. Our equation would then say Y equal 80 / X and we want to find the Y when X is 2, so we're going to substitute in two and we're going to get Y equal 40 when X is 2, a joint variation if Y varies jointly as X and the square of Z and inversely to West. So we're going to multiply by K the constant, and then we're going to say times X * Z ^2 all divided by W Let's say y = 6, W equal 4X equal 2, and Z equal 1. So we can then find the constant. We just literally fill in those numbers. So if we want to solve this, we'd get 24 equaling 2K or K equal 12. So we'd have Y equal 12 XZ squared over W. Thank you and have a wonderful day.