Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Word problems with two variables. We have 5 steps whenever we do any word problem in mathematics. These are pretty standard steps. We want to define our variables. We want to set up the equations, solve the equations, check your solutions, and answer what is asked for. So the first problem says a customer walks into an electric store, electronic store and buys five MP3 players and eight sets of headphones, paying $840. A second customer buys three MP3 players and four sets of headphones and pays 480. How much does an MP3 player cost and how much does a set of headphones cost? So we're going to let X be the quantity or the amount of MP3 players and Y be the quantity of headphones. Now looking at these prices, my guess is this is pretty dated problem because many years ago these were more expensive. So we have five MP3 players or 5X's and eight headphones, 8Y equaling 840. That's our first equation. 5MP3 players, remember the MP3 player was X8 headphones, that's our Y and eight 40s our total. Then our second customer did three MP3 players or 3X and four sets of headphones for 480. When we want to solve this, we're going to take the second equation and multiply it all the way through by a -2. So I'm going to have five X + 8 Y equaling 8:40. But when I take that second equation and multiply it all the way through by a -2 negative six X -, 8 Y equaling -960. Now the reason to do the -2 is so that we can get additive inverses on our YS. The +8 and -8 Y are going to cancel, so we get negative X equaling -120 or X = 120. If X is 120, we're going to then substitute it back in to find our Y, and it doesn't matter which equation we substitute it back into. So let's say 3 * 120 + 4 Y equal 480. So 360 + 4 Y equal 484 Y is going to equal 120, or Y is going to equal 30. Now we need to check in both equations. Just because it's true in one doesn't necessarily automatically tell us it's true in the other. So 5 * 120 + 8 * 30 equal 845 * 120 is going to be 600 + 8 * 30 is 240 equaling 840. And we can see that one's true when we check in the other one, 3 * 120 + 4 * 30 equaling 480, well, that's 360 + 120 and we can see that that one is also true. So our five steps, we defined our variables, we set up our equations, we solved our equations, we checked it. And so our last step is to answer the question, how much does an MP3MP3 player cost and how much does a set of headphones cost? So the MP3 player costs $120.00 and the headphones cost 30. Now there's many ways to do mathematics, but that's one method that's fairly simplistic to get to, or fairly easy. We could have come back here and we could have eliminated the X's instead of the Y's instead of doing a substitution. More than one way to do math. Walt made an extra 9000 last year from a part time job. He invested part of the money at 9% and the rest at 8%. He made a total of $770 in interest. How much was invested at 8%? So let's let X be the amount invested at 8%. Actually, let's doesn't matter, let's make it 9% just so it's in the same order as what was given. So Y equals the amount invested at the 8%. Nine percent came first in the story prompt. Together he invested $9000. He had some of it in nine percent, so point O 9X, that's the amount of interest he made on the money. He put in the 9% point O8Y for the amount of interest he made at the 8%, and he made a total of $770. Well, the first thing I'm probably going to do is get rid of those decimals. So I'm going to multiply everything through by 100. So I get 9X plus 8Y equals 77000. Now I want to know how much was invested at 8%. So I actually at this point don't really care about how much was invested at 9%. So I want to get rid of the X because I want to solve for Y. So I'm going to take this top equation and I'm going to multiply that top equation all the way through by a -9 so that I can get rid of the X's so that when I add these two equations together, I get negative Y equaling negative. Well, there's going to be 3 zeros at the end. 77 from 81 is going to give us four. So the Y had 4000. Now technically that's all I need because that's how much was invested at 8000 or 8%. But we really need to figure out the Y and be able to check it. So if I know the Yi can substitute it back into either of the two equations, obviously substituting it into that first one is going to be easier. X + 4000 is 9000, so Y has got to be 5000. So from here, we're going to check it over here. We're going to check. So remember there are five steps to a story problem. The first one over here is to define the variables. The second was to set up the equation. The third was all of this portion, which was to solve it. The 4th is to check. So does 5000 + 4000 equal 9000? That one does check. We have to check them in both because just because it works in one doesn't necessarily mean it works in the other. I always recommend to students to actually to anybody to check it in the original. I check it with the right numbers though. That will help because if you made any kind of a computational error between the setup of the equation and the actual using the equations, it could check in a piece further along, but maybe not in the beginning. So I always, always recommend to check in the very very beginning. So when we look at this, 1.09 * 4002 decimal places will take off 2 zeros at the end. So 4 * 9 is 3/6 and we have 10 left here. 2 decimal places will take off 2 zeros at the end. So we really end up with 8 * 50. So 8 * 5 is 40 with a zero at the end. So uh oh, 36 nine percent, 8 percent, 5004 oh, this was X, this wasn't Y Ah, see, my check didn't work and now I can go back and figure out why it didn't work because X was 5 and Y was four. So a moment ago I really was doing it right and I didn't realize it. So let's go back here and do this again. So .09 * 5000 is going to be 450 and .08 * 4000 is going to be 320. And now we can see that that one really does work. So how much was invested at 8 percent, $4000 invested at 8%. Looking at another example, Ziggy's famous yogurt blends yogurt regular yogurt that's 3% fat with its non fat yogurt to obtain low fat yogurt that is 1%. So if we let X be amount of of 3% and Y equal amount of no fat, we can see that the quantities X + y is going to equal a total of 60 lbs of low fat yogurt. Now the percentage .03 X no fat yogurt has 0% fat in it, so it's going to be 0 Y. So percent times amount plus percent times amount is going to equal our percent, in this case 1% times our amount of 60. So the second equation is actually not going to have any YS in it. We're going to get .03 X equaling point 0. Nope, .6. So if we divide each side by .03, we're going to get .6 over point O3. If I multiply the top and bottom each by 100, I'd get 60 / 3, or 20 for my X. If I have 20 for my X, I'm going to put that in. For my Y, I'm going to put it into the other equation to solve for the Y. So 20 + y = 60. So Y is 40. So our first step define our variables. Our second step set up our equations. Our third step all of this was to solve the equations. Our 4th step is to check. So does 20 + 40 = 60 and the answer is yes. But then we need to check in the other equation also. So .03 * 20 + 0 * 40. Does that equal .01 * 60? Well here we have two decimal places, so we're going to get .6 plus 0 equaling .6. So that one really does work. And so our last step is to answer what was asked for. How many pounds of non fat yogurt should be mixed to obtain 60 lbs? So we need the non fat yogurt. So we need the Y value which was 40 lbs of non fat yogurt, no fat yogurt. How many pounds of no fat yogurt should be mixed to obtain 60 lbs of low fat? So it didn't ask us how much of the 3%. The next problem in a chemistry class, 12 liters of a 12% alcohol solution must be mixed with a 20% solution. To get a 14% solution, how many liters of the 20% solution are needed? So we know 12 liters of a 12% must be mixed with a 20%. We don't know how much of the 20%. So let's X equal amount of 20% and to get a 14% solution. So Y is going to be the amount of the 14% solution. So we have 12 + X which is going to give us our Y 12 liters of a 12% must be mixed with 20% to get 14%. So now we're going to have our second equation being the percent times the amount plus the percent times the amount equaling the percent times the amount. Some things that sometimes I have found useful is to realize that my percent of my mixture, this 14, has to be in between the percents of each of the two things I'm combining. So 12 and 20. If I had had the 20 over here, I couldn't combine a 12% and a 14% to get a higher percentage. So this percentage over here, the mixture has to always be in between the other two. The other thing to think about is that we could actually multiply it all through by 100 like we did before. So we could get 144 + 20 X equaling 14 Y. Now I like to keep my numbers as small as possible and I can see that all of those are divisible by two. So we could actually take all this equation and divide it by two to get a new equivalent equation. So 72 + 10 X equals 7 Y. Now up here, this is already solved for Y, so I'm just going to substitute it in on this other side to solve it. Instead of doing elimination, I'm going to do substitution. So 72 + 10 X equals 7 Y. But Y was 12 + X, so 72 + 10 X equal 84 plus 7X. Taking all the XS to one side and everything else to the other, I can see that X is going to equal 4. If X is 412 + 4 would give me 16 for my Y. We need to check them. So does 12 + 4 give me 16? Yes, but we also need to check it in that other equation .12 * 12 plus .20 * 4. Does that equal .14 * 16? Well, this is going to be .12 * 12 is going to be 1.44 point 2 * 4 is going to give me .8 and .14 * 1 * 16 is going to give me 2.24. So when we add these together, we can see that those really do check when we come back to the question how many liters of the 20% are needed. So our answers the X. So we need 4 liters of 20% solution. Our next example, a boat traveled 168 miles downstream and back. The trip downstream took seven hours. The trip back took 42 hours. Find the speed of the boat in Stillwater and the speed of the current. So we have rate times time equaling distance. Sometimes we actually will put this one in a box just to kind of help keep our information organized. So it tells us our distance downstream and back is 168 one 68168. The trip downstream took seven hours, so 7 hours for down and up. The trip back took 42 hours. That's a long trip back. Find the speed of the boat in Stillwater and the speed of the current. So let's let X be the speed of the boat and C be the current. If we're going downstream, the current is making us go faster. So X equals speed of boat and C equal current. Now, the easiest way I can tell you to think about this is the speed of the boat is always, always, always going to come first. And the current affects the speed. So if we're going downstream, the currents pushing us along or helping us go faster, we don't have to do as much if we're going downstream. If we're going upstream, we're working against the current, IE it's going to be a subtraction. It's much harder to go upstream than downstream. So now we're going to have our two equations, our rate X + C * 7. Our time is going to equal 168 and our X -, C * 42 is going to equal 168. Now I'm always into trying to get easy things. I'm going to see is 168 divided by divisible evenly by 7 and the answer is yes. If I divide each side by 7, I get X + X + C equaling 24. Now there's more than one way to do it. I could have distributed the sevens out, but I like easy numbers if I can. Now I'm going to see is the 168 divisible by 42? And guess what? The answer there is also yes. So X -, C is going to equal 4. So these two new equations I can add exactly like they are because the C's are additive inverses right now. So we get 2X equaling 28. So X equal 14. If I can find XI should be able to find my C So 14 + C is 24 C equal 10. So my my five steps, I've done three of them. I've defined my variable up here. I've set up my two equations and I've solved those equations. You can hear my cat in the background. Sorry about that. She's noisy. So our 4th step is to check and I always suggest checking in the originals. So does 14 + 10 times 7 equal 168? Well, 14 + 10 is 2424 * 7. Seven times 4 is 28, so we're going to carry two 7 * 2 is 14 plus the two is 168. That one definitely checks. If we look at the other one, we get 14 -, 10 times 42 and does that one equal 168? So we get 4 * 42 and 4 * 42 is 168 = 168 and that one also checks. So our question said find the speed of the boat in Stillwater and the speed of the current. So our last step is to answer what's asked for. The speed of the boat in Stillwater is that was my X and my X was 14 and our units are mph and the current is 10 mph. Thank you and have a wonderful.