Hello wonderful mathematics people. I'm Anna Cox from Kella Community College. Using elimination and substitution to solve word problems. There are five steps in a word problem. You should always define your variables, set up the equations, solve the equations, check your solutions, and answer what is asked for. If a problem is given to you in words, you should always answer in words. A customer walks into an 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? How much does a set of headphones cost? So our first thing is going to be to define our variables. Let's A equal cost of MP3 player and B be the cost of headphones. So our first sentence says we bought five MP3 players and eight headphones for a total of $840. So the amount times the cost plus the amount times the cost equal the total amount. So the amount there were five of them times the cost we don't know, plus there were eight of them times the cost of the headphones we don't know equal the total cost. The second one says there were three MP3 players and four headsets and that total was 480. So once again, the quantity times the cost. The quantity times the cost equal the total cost. Now we need to solve these. We need to get either the coefficients on the A's or the B's to cancel. The B's are a little easier because if I took the second equation and multiplied it by -2, I can get the coefficients on the two equations to be the additive inverses. So if I multiply the second one by -2, I'd get -6 A -8 B equal -960. If I looked at then adding these two equations 5A and -6 A is going to be negative 1A my 8B and -8 B are going to cancel 840 and -960 is going to give me a -120. To get A by itself, we're just going to divide each side by a -1 and we're going to get a equaling 120. If I can solve for one of the variables, then we can substitute back to solve for the other. Doesn't matter which equation we put it in. So 5 * 120 + 8 B equal 840. Well, 5 * 120 is 600 + 8 B equal 840. If we subtract the 600 we get 8B equal 240. If we divide each side by 8, we get B equal 30. So we've solved the two equations. Now to check them we need to figure out does 5 * 120 + 8 * 30 really equal 840? Well, 5 * 120 is 608 * 30 is 240. So this one checks. Now it's important that you check both equations and the reason is is it might check in one but not in the other. So 3 * 120 + 4 * 30 does that equal 483 * 120 is 364 * 30 is 120. So 480 does equal 480. The last step in a story problem is to answer what was asked for. Well, what was asked for. How much does an MP3 player cost? So the cost of an MP3 player is $120.00 and the cost of the headphone is $30. 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 in a story problem, our first step should be to define the variables. Let X be amount invested at 9% and Y be the amount invested at 8%. So we know the amounts invested at 9% and at 8% was a total of $9000. We also know that the money invested at 9% gave us a nine percent return or .09 times the amount of money that was invested there. The amount of money invested at 8% gave us an 8% return, or .08 times the actual money invested there. The total interest was $770. Now to solve this, we're going to try to get the coefficients on one or the other variable to be the additive inverse. So we might take this top equation and multiply it by -.08. So we'd get negative .0 eight X -, .08 Y equaling -720. We're going to leave the other equation exactly the same, so we get .0 nine X + .0 eight y = 770. When we add these together, we get .01 X equaling 50. Divide each side by .01 and we get X equal 5000. If we know X is 5000, we can then substitute it into the other one of the other equations to figure out our Y. So 5000 + y equaled 9000 or Y equal 4000. So now what we're going to do is we're going to check both equations and we have to check them both because it may work in one but not in the other. So 5000 + 4000, does that really equal 9000? Yes, 9% * 5000 plus 8% * 4000. Does that equal 770? Well, 9% of 5000 is 458% of 4000 is 320 and 450 + 320 is 770. So what did it ask? It asked us how much was invested at 8%. So the amount invested at 8% is $4000. Ziggy's famous yogurt blends regular yogurt that is 3% with its non fat yogurt to obtain low fat yogurt that is 1%. So let's let X be the amount of regular yogurt and Y be the amount of no fat yogurt. It says how many pounds of regular yogurt and no fat yogurt should be mixed to obtain 60 lbs of low fat yogurt. Well, X + y is going to equal 60. The amount plus the amount equals the total amount. Now we know that the regular yogurt is 3% point 03 X and we know that non fat yogurt has how much fat? Well if it's non fat it has 0% and we want it to mix to give us 1% of the entire 60 lbs. So this is a percent times amount plus percent times amount equal percent times amount. To solve this we actually want to look at this bottom equation first because we know that .03 X is going to actually equal .6 because the 0Y canceled. So now if we divide each side by .03, we're going to get X equaling 20 if X = 20. Now we can do a straight substitution into the other equation to figure out our Y. Our Y is going to equal 40. So now we need to do a check. The check would be does 20 + 40 really give us 60? And the answer is yes. And does .03 * 20 plus 0 * 40 equal .01 * 60? Well .03 * 20 is .60 times anything is 0.01 * 60 is .6 so it checks. The last part is how to answer what's asked for how many pounds of regular yogurt and how many pounds of no fat yogurt. So we need 20 lbs of regular yogurt and 40 lbs of non fat yogurt to get 60 lbs of low fat yogurt in a chemistry class. 12 liters of a 12% alcohol solution must be mixed with 20% solution to get a 14% solution. How many liters of the 20% solution are needed? Let X equal the amount of the 20% solution and Y equal the amount of the 14% solution. So 12 liters plus the amount of the 20% is going to equal the amount of the 14%. The other equation's going to take into account the percentage times the amount. So 12% times the 12 liters plus the percentage. In this case, 20% times the amount is going to equal percentage times the amount. Now there's some helpful hints here. If I have a 12% and a 20% and I'm mixing them together, it should make sense that 14% would have to be in between 12 and 20. If we look at this problem, the Y is already solved in the first one, so we could actually do a straight substitution. 12 * 12% is going to be 1.44 + .20 X equal .14. Instead of the Y, we could substitute 12 + X. Now if we solve this equation, because we have only one variable in it, .14 * 12 is 1.68 and .14 * X is .14 X take all the XS to one side and everything else to the other and then divide. So we should get X equal 4. If we know X we can find Y just by substituting. So 12 + 4 would give me my Y of 16. Now we need to check both equations. Just because it works in one doesn't necessarily mean it'll work in both. So does 12 + 4 = 16? Yes. How about does .12 * 12 + .20 * 4 equal .14 * 16? Well, .12 * 12 is 1.44 point, 2O times 4 is .8 and .14 * 16 is 2.24. So it does indeed check. So what did it ask for it? It asked for how many liters of the 20% solution are needed. So we need 4 liters of the 20% solution. 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 the first thing we need to realize is that rate times time equal distance and we have two unknown rates. We have the rate of the boat in Stillwater and we also have the rate of the current. So if we're going with the current, IE downstream, we'd have X plus the current times the amount of time it took, which was 7 hours to go downstream to get the total distance of 168. When we're going upstream, we're fighting the current, so we would subtract the current. It took 42 hours to go the 168. Now let's let X equal the speed of the boat in Stillwater and C be the speed of current. Now that we have our two equations, let's go ahead and distribute. So we're going to get seven X + 7 C equal 168 and also 40 two X -, 42 C equal 168. To solve this by elimination, we're going to take this top equation and multiply it by 6. So 40 two X + 42 C equal 1008. We're going to leave the bottom equation the way it is so that the CS are additive inverses, and when we add them, they'll cancel. So we're going to get 84 X equaling 1176. If we divide each side by 84, we're going to get X equal 14. Once we find X, we can substitute it into either of the two equations to get the C 7 * 14 is 98. If we subtract 98 from each side, we get seven C = 70 or C equal 10. We have to check them. So does 14 + 10 * 7 equal 168? Well, 14 * 10 is 24, and 24 * 7 is indeed 168. Just because it checks in one doesn't necessarily guarantee it checks in the other. 14 -. 10 * 42 equal 168 four times 42 does also check. The final thing is to answer what's asked for. So the speed of the boat in Stillwater is 14 mph and the current is 10 mph. Thank you and have a wonderful day. This is Anna Cox.