Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College solving formulas and direct and inverse variations with joint variation. So we have an equation that's given to us as PV divided by T equal K and we want to now solve for T So we always want to multiply and get rid of the denominators. So our common denominator here is T. If we multiply each side by T, the TS on the left side are going to cancel because one's in the denominator and one's in the numerator, and we're going to get PV equal TK to get T by itself. Then we're going to divide each side by K. So T is going to equal PV divided by K This next one we don't want to solve for N so we're going to start by multiplying by the common denominator, which in this case is M plus PN. And if I multiply 1 side by the common denominator, we have to multiply the other side by the common denominator. On the right hand side, the M plus PN and the M plus PN are going to cancel. On the left hand side, we're going to distribute IM plus IPN equal PT. We want to solve for N so we need to get every term without an N to the other side. So we're going to subtract. And then to get N by itself, we're going to divide each side by IP, so PT minus IM over IP. Now there are more than one way to do this. One, once we had gotten rid of our denominator before we distributed things out, we could have divided each side by I. So M plus PN would equal PT divided by I. Then we would have to subtract the M. And finally, to get the N by itself, we would divide each terms by P and then the P and the P could cancel so we get TI minus MP. So more than one way to do this. Let's look at some more. This isn't on a formula where we want to solve for S So if we have F equaling SG over s + v, our first thing is going to be to multiply by the common denominator, which is s + v And if we multiply 1 side by the common denominator, we have to multiply the other side by the common denominator. The s + V's on the right side will cancel. On the left side, we're going to distribute. So now we want to solve for S We need to get all the s s on one side. Once the s s are on one side, there's multiple terms, so we're going to factor the S out. To get S by itself. We're going to divide each side by g -, F If I divide one side, we have to divide the other, and that's our answer because the g -, F on the right will cancel. If we look at this next one, the formula involves the work total T for some work to be done for by two workers who individual times are 1 / A and 1 / B equaling 1 / t So the common denominator here is going to be a BT. If we multiply everything through by the common denominator, we're going to get ABT times 1 / A plus abt times 1 / B equaling abt times 1 / t The A's are going to cancel in the first term, the B's are going to cancel in the second term, and in the last term the T's are going to cancel. We want to solve for T, so T terms are currently already on one side by themselves, so we're going to factor out the T, and our final step is to divide each side to get the T by itself. And we usually list them in alphabetical order, A+B. Looking at this next one, we want to get D1. So the common denominator is going to be W2D2. The subscripts just mean that they're different works in different distance. So we're going to have W2D2 multiplied by each side. On the left side, the W twos are going to cancel and on the right side the D twos are going to cancel. So we're going to get D2 W one equaling W2D1, and we want to get D1 by itself, so we're going to divide each side by W2. Looking at this next one, it's going to be really similar to the work rate one. Our common denominator here is going to be capital R, little R1 and little R2. If we multiply everything through by capital R, little R1, little R2, we're going to get things to cancel and simplify. So the capital Rs are going to cancel, leaving little R1, little R2 equaling. The R ones are going to cancel, leaving capital R little R2, and then the R twos are going to cancel, leaving capital RR 1. The capital R currently is all on one side, so we're going to factor it out to get it by itself. We're going to divide by that R2 plus R1. I'm going to put the R1 first because in numerical order we usually go smallest to largest. This next one we want to solve for V1. So we're going to start by getting rid of the denominator 2S equal V1 plus V 2 * t This one, we have a couple different ways to do it. If we are consistent with what we've been doing, we would distribute the T and then we want V1. So we're going to take that V2T over to the other side and to get the T by or to get the V1 by itself, we're then going to divide by the T. That's a perfectly acceptable answer. A different method would be when we had this two SV, one plus V 2 * T to divide each side by T, and that would give me V1 plus V2. Now when I want the V1 by itself, I would subtract V2 from each side. Those are the same answers. They look a little different, but they really are the same. If you simplify this one so that it's separate terms, you'll see that if we go to this next one, we're going to multiply each side by v + 2 R, So IV plus 2R on the right hand side, the V + 2 Rs are going to cancel. We're going to distribute this out. We need to get everything with AV on one side, so we're going to subtract. Once the V's are all on one side, we have to factor it out. And then to get V by itself, we're going to divide. This one's going to be very similar to the one we did earlier. I think I let you do that one on your own. This one is a little different. The common denominator here is going to be AB. So we're going to have AB times t / a plus AB times t / b equal AB times 1. So we're going to get TB plus TA equal AB. We're solving for B, so we need to subtract that BT term or TB term. We can multiply in any order we want. We're going to factor out AB because we're trying to solve for the B. So our final step is going to be to divide by a -, t If we look at this next one, we're going to multiply to get rid of the denominator I * e plus NR equal NE. We're going to distribute. We need to get all the ends on one side, so we're going to subtract. Once all the terms with N are on one side, we're going to factor it out. And then finally we're going to divide, IE divided by E minus IR equal N. The same thing on this one. We're going to multiply to get rid of the common denominator here. We're going to distribute. We're going to take every term without AT 1 to one side, and then to get the T1 by itself, we're going to divide. Now this is one where we could have divided by SM in the beginning. So we could have had our T 1 -, t two equaling H divided by SM. Then T1 would be H divided by SM plus T2. Multiple ways to do that, not on all of them, but on some of them. Here we want to solve for little E. So if we multiply by the common denominator and then we want to divide by the R Plus capital R Plus little R. So we get ER over capital R Plus little R equal little EI got rid of it by multiplying by the common denominator and then solve for my variable that I was trying to get alone here. If we're trying to solve for R, if we start by distributing, subtract so all the terms without an R go to one side and then divide so that any term without the R is on the left or on one side. This one is one of them that we could have started by dividing each side by the P. So a / P would equal 1 plus RT if we subtracted the one. And then to get the R by itself we would divide each term by T. Those two answers truly are equivalent. If you don't see it, take some time and practice separating the two terms and making them simplified. I've now listed the answers. OK, now we're going to move into variation. Variation is a relationship. If we have a direct variation, it's going to be the form Y equal. MXM is some constant. Many books have it as Y equal, KXK being a constant. Now, this is a formula that hopefully we all recognize, because that's really really just the equation of a line with no Y intercept. If there's no Y intercept, it means that the Y intercept is understood to be 0. So it's really going through the origin. So this is just the equation of a line. So this first one says Y equal 54 when X equal 12. And we want to figure out that constant or that M or that K. So to get K by itself, we're going to divide by 12 and we get 54 twelfths equal K. If we reduce that, we'd get 27 sixths, which can still be reduced, so we'd get 9 halves. So the equation would be Y equal 9 halves X. The constant of variation is 9 halves. The equation is Y equal 9 halves X. This next one Y is 4 when X is 30, so 4 thirtieths equals our K. If we reduce that we get our constant being 2 fifteenths. So our equation is Y equal 2 fifteenths X. If we look at the next 1.9 equal K * .4 get K by itself .9 / .4. We don't leave decimals inside of fractions. So multiply the top and the bottom each by 10 and we get our constant being 9 fourths. So our equation would be Y equal 9 fourths X. Now we can do some word problems. The number of aluminum cans used each year varies directly as the number of people using the cans. So we could say that N / P equal K So if 250 people used 60,000 cans in one year, how many cans are used each year in Dallas? If there's one, how many cans? Yep, N is the number cans in the population is 118-9000. So what we would do is we would cross multiply and divide, and I don't know off the top of my head that number, but you can grab your calculators and compute it. That's going to be a large number of cans because of 250 people use 60,000 cans, 1,189,000 people are going to use a lot of cans. So compute that out. Put the correct answer on your worksheet. The average US community of population 12,500 released about 385 tons of lead into the environment. How many tons were released nationally? So population equal K times the tons. So population over the tonnage equal the constant. So if there was 12,500 population released 385 tons. If we have 281,000,000 population, how many tons? Here let's multiply by our common denominator. So 12,500 T is going to equal 385 * 2 eight one with six zeros at the end. To get T by itself, we're going to divide. Once again, grab your handy dandy calculator, compute that, and write it down. The maximum number of grams of fat that should be in a diet varies directly as a person's weight, so fat overweight equal our constant. If a person who's weighing 120 lbs has 60 grams of fat per day, what is the maximum daily intake for a person who weighs 180 lbs? We're going to cross multiply, so 60 * 180 equal 120 * F Divide each side by 1:20 and we're going to get F equaling 90, so 90 grams. The number of kilograms West of water in a human body varies directly as the mass, so water to mass equal your constant. So a 96 kilogram person contains 64kg of water. How much water does a 60 kilogram person? So I'm just setting it up. Water to mass equal water to mass. If we cross multiply to get the West by itself, we then have to divide by 96. And please compute that out. The weight M of an object varies directly. The weight M of an object on Mars varies directly as its weight on Earth. So Mars divided by Earth is going to equal our constant. A person who weighs £95 on Earth weighs 39 lbs on Mars. How much would 100 LB person that's assuming we're on Earth weigh on Mars? We're going to cross multiply and then divide by the 95 inverse variation. So this time Y is going to be K times the inverse of the variable, or K * 1 / X. So if 14 = K * 1 / 7, to get K by itself, we're going to multiply each side by 7, so K is going to be 98. You could also think of that K as K in the numerator and seven in the denominator. Some students like that better, they think it's easier to see. So the next problem we have Y equal 1 when X = 8, so our K equal 8. So our equation is going to be Y equal 8 / X. If we go back to the last one, our equation is Y equal 98 / X, so Y is 12 when X is five. Solving for K, we're going to multiply each side by 5. So our equation is y = 60 / 5. The time T required to do a job varies inversely as the number of people working. So T the time times the population is going to equal the constant. So the time was five hours for seven brick layers equal how long? So T * 10 brick layers. So T is going to be 5 * 7 / 10 or T is going to be 7 halves, so 3 1/2 hours. If we look at the next one, musical pitch, the pitch of a musical tone varies inversely as the wavelength. So pitch times wavelength equals our constant. So 330 * 3.2 is going to equal 550 times. Let's see wavelength. So now we're going to divide 330 * 3.2 / 550 equal RW. Go ahead and grab the calculator and compute that this next one. The wavelength of a radio varies inversely as the frequency. So wavelength times frequency would equal our constant. So if we have a wave of 1200 and a frequency of 300, we equal 800. That's the frequency. So we need the wave. So to get the West, we're just going to divide by 800. The next problem, the weight that a horizontal beam can support varies inversely as the length. So the constant is weight times length. 8 meters times 1200 grams is going to equal 14 meters times how many grams. To get the grams by itself, we're going to divide by 14. Rate of travel T required to drive a fixed distance varies inversely as the speed, so time times the speed is our constant. So five hours times 80 is going to equal time times 70. So time is going to be 5 * 80 / 70 and you're going to simplify those up. Now we also have joint variation, and joint variation means that things are happening more than once. So if we have a joint variation problem, it might be something like Y varies directly as the square of X and inversely as Z. Then it would probably give us information like when Y equal 100, X equal 5, and Z equal 4. So our equation would be Y equal K If it varies directly as the square, we're going to say X ^2 inversely means that it's 1 / Z. Or we could think of that as KX squared in the top with Z in the bottom. Now we just plug in our numbers 100 equal K, 5 ^2 / 4 four 100 equal 25 K so K is 400 / 25 or K = 16. So our equation would be y = 16 X squared over Z. Thank you and have a wonderful day.