Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College solving formulas and variations. The Doppler effect is given by the formula F equal SG divided by the quantity s + v We're going to want to solve for S so that's the sound in a particular medium. So to solve for S, the first thing we're going to do is we're going to multiply by the common denominator, which is S + V So I get F * S + V on one side, and the S plus VS on the other side are actually going to cancel. So S + V cancels. So if I distribute, I get FS plus FV equaling SG. But I want to solve for S, so I need to get all the s s on one side. So we're going to subtract the FS so that all the s, s are one side. The next step would be to factor out the S since it's in multiple terms and to get the S by itself. The last step would be to divide by whatever was being multiplied by our S, So in this case, the g -, F Our next example is a work formula and we're going to want to solve for T We could solve for A or B also. So once again, our common denominator in this case, it's going to be AB and T. So we're going to have ABT times 1 / A plus ABT times 1 / B equaling ABT times 1 / t So the A's cancelling, we get BT plus AT equaling AB. Our goal again to solve for T. So we're going to get all the terms with T on one side and factor it out so that we have T times something. And to get the T by itself, we're just going to divide each side by b + a. So for all of these examples, we're getting the denominator to go away first. So I * e plus NR and that's going to equal the Northeast. Then we want to solve for N so we're going to distribute. Once we have it distributed, then we're going to take all of the things with an N in it to one side. Once I have all my terms with an N in it on one side, then we're going to factor out the N. And our last step will be to divide, so IE divided by E minus IR all equals the N1. More, when we solve for V we get I * V + 2 R equaling 2V. So distribute, look for what we're solving for and get all those terms to one side. Once all the terms with the variable we're solving for on one side, we factor out the term, this case the V. And then our final step is to divide. Left you several to try. We're going to then talk about variations. There are three different types of variations. We're going to study direct variation where we find the variation, constant and equation of variation in which Y varies directly as X. So in variation we have a constant. So direct variation says we have Y equaling some constant directly times the X. So 54 = K * 12. So we can actually find that variation constant, it's 54 / 12. If we reduce that, we'd get three goes into both no? Yes, three does go into both 18 / 4 or 9 / 2. So our variation constant is 9 halves, but our equation of variation is Y equal 9 halves X because once we find that K that K is always a constant no matter what the parameters. For this particular problem, the number north of aluminums cans each year. So N varies directly as the number of people using the cans. So if 250 people use 60,000 cans, so 60,000 cans for 250 people, so we can solve our K to get 60,000 / 250. And then it wants us to actually know how many cans are used each year in Dallas. So our equation would be N equaling. I don't know what 60,000 / 250. Let's see, we take off a zero from the top and then that would give me 60,000 / 25. We're actually we're down to 6000, aren't we? Take a 5 out of each 121200 over 5 S 240. So our K is 240. So now we want to know how many cans the number of cans are used for a population of 1189000. I don't know that off the top of my head. Go ahead and multiply that out. And that's how many cans are used With this varying directly. Inverse variation is similar, but now instead of varying directly, we're going to have the inverse of X. So Y equal K * 1 / X. Or sometimes it's thought of as Y equal K / X. So if 14 = K / 7, we can solve for AK and get 98. So our equation would be Y equal 98 / 7 or by X so Y equal 98 / X. So if X was seven, our Y is 14. Work rate. The time required to do a job T varies inversely as the number of people working on it. So K / P So it takes five hours for seven brick layers. So we can see that K is 35. How long will it take 10 brick layers? So time equal 35 / 10. So time would be 3.5 hours, and that makes sense. If it takes 7 brick layers 5 hours, 10 brick layers should take less, so 3.5. Joint variation occurs when there's multiple things going on. So find the variation constant when an equation of variation and an equation of variation when Y varies directly as the square of X and inversely as Z. So sometimes I just write my equation as I'm going and then I can simplify it. So this is really going to be Y equal KX squared over Z. So when Y is 100, we don't know K. We know X is five and we know Z is 4. So we get 400 equaling 25K. So K is going to equal 16. So our equation is going to be y = 16 X squared divided by Z. Thank you and have a wonderful day.