Direct Variation Applications - Today, we'll be looking at direct variation. And our goal is to solve direct variation problems. The variable y varies directly as x, but there is some positive constant k, such that y equals k times x. We also say that y is directly proportional to x. And k is called our constant of variation. My connection I would like to make, direct variation equation can be written as y equals mx, instead of y equals kx. So you can almost think of variation as a linear equation or the y-intercept is 0. And the slope would be a constant of variation. And I'm going to highlight some things as we read. The weight, M, in pounds of an object on the moon is directly proportional to the weight of that object on Earth. An astronaut who weighs 140 pounds on Earth will weigh 22.4 pounds on the moon. Find an equation of variation. So our goal is to find the equation that represents the relationship between M and E. Now going back to our notes, we know that the variation equation is y equals kx, but on this problem, we're not using y and x. We're using M and E. That's really the only difference. So the weight, M, is directly proportional to E. The way you write that would be M is equal to k times E. This is our general equation, but we want to find the value of k to make it the equation of variation. And we can do that by using the next sentence. An astronaut who weighs 140 pounds on Earth-- so when E equals 140 pounds, they'll weigh 22.4 pounds on the moon. M is equal to 22.4. So, again, when E is 140, M is 22.4. We can use these values in this equation and find our content of variation. Let's go ahead and do that. 22.4 is equal to k times 140. So to solve for k, we'll divide by 140. You can check me on this, but k should equal 0.16, which would give us our equation of variation as the weight on the moon is equal to 0.16 times the weight on Earth. This is what we call our equation of variation. A constant of variation would be 0.16. We have one more question. An astronaut who weighs 120 pounds on Earth, how much will the astronaut weigh on the moon? So the Earth weight or E is now 120 pounds. Going back to our previous problem, we know that our equation is M equals 0.16 E. So in order to find the weight on the moon is as simple as doing a little substitution, 0.16 times 120. And you can check my math, but I believe that's 19.2 pounds. So someone that weighs 220 pounds on the Earth will weigh 19.2 pounds on the moon. I hope that helps. And I hope you have a good day.