Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Word problems with two variables. When we do a word problem there are five steps we should keep in mind. We should always define our variables so that we know what our unknowns represent. Setting up the equations and then solving the equation. Check your solution. Make sure your solution works in both the original equations and then actually answer what is asked for. If a problem is given in words, typically we give our answer in words. So the first example, a customer walks into an electric store and buys five MP3 players and eight sets of headphones paying $840.00. So let's let A equal number of MP3 players and B equal number of headphones. So I've just done the first step. I've defined the variables so we know that five MP3 players and eight headphones is $840. There's my first equation, five of the players and eight of the headphones equal 840. Our second statement says a second customer buys three of the players, so 3A and four of the headphones spending $480. So in order to solve this, this was our step one. Remember we defined our variables. Step two was set up the equations. Step three is to solve. So when I do this, I'm going to use the elimination property and I want to get the coefficients on the B in this case to be the same. It's the easiest to be the B because we want additive inverses. So if I multiply the second equation through by negative 28B and -8 B would cancel leaving me just A's and a constant. So if I have 5A plus 8B equaling 840 and if I multiply that second one by -2 negative six a -, 8 B equal -960 so that when I add those straight down I get a negative a equaling -120. So A is going to be 120, and if we know A, we can put it in either of the two equations, or we could do elimination again. I think maybe I'll actually do elimination again just to practice it. So if I'm going to eliminate the A's this time, I'm going to multiply the top equation by three and the bottom equation by a -5 S three is going to give me 15. A plus 24B equal 02 252O and if I do the bottom one by a -5 I get -15 A -20 B equal 004, negative 24. So when I add those we get 4B equaling 120 so B is 30. So that whole thing was Step 3. Step 4 is to check it and I have to check it in both. Does 5 * 120 + 8 * 30 equal 840 and the answer is yes, we get 600 + 240 and that is 8:40. So that's check one. Then we need to know does 3 * 120 + 4 * 30 equal 480? And the answer for that one is yes also. So it wants to know how much does an MP3 player cost and how much does a headphone cost. So it actually wants both. So the MP3 player costs $120.00 and the headphone cost 30. That was my last step. That's by Step 5. So when you're doing this story problems, try to make yourself identify all five steps and make sure you've done them all. This next example. 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 X be amount invested at 9% and Y be the amount invested at 8%. So we know the total money he made last year would be the amounts that he invested or X + Y is 9000. Now that's the quantity of money he actually had. But we want to figure out the interest on the money. S percent times amount plus percent times amount is going to give us the total interest earned because the entrance is a percent times an amount. It's a total of a percent times amount, the 9000 times whatever that interest rate would have been as a accumulation. So to solve this one, the first thing I might point out is we can multiply everything through by 100 in the second equation. Most students, most people in general, actually don't like to deal with decimals if they don't have to. So when I multiply that by 100, I can see now that if I take this top equation and multiply it by a -9, so I get -9 X -9 Y equaling -8100 so that when I add those I get a negative Y equaling -4000 or Y is 4000. If I wanted to do the same thing, I could take that top equation and multiply through by a -8, so I'd get a -8 X -8 Y equaling -7200 to combine with my nine X + 8 Y equaling 77000. When I add those, I get a single X equaling 5000. So our first step was to define the variables. Our second step was to give the equations. Our third step, this whole thing of solving them. Our fourth steps to check does 5000 + 4000 equal 9000 and the answer is yes. And also does 9% of 5000 + 8% of 4000 equals 770 and if we check that the answer is yes there. Now this time it is actually asking us only how much was invested at 8%, so the amount invested at 8% is my Y values so 4000. I should mention that when you do your check, you always do your check in the original equations. Because if I made any kind of an error between the original equation and the actual computation, if I stick it into one of these other equations later on, it might check where it wouldn't check necessarily in the beginning. So let's look at a couple more and I think maybe I'll set them up and let you solve them. So Zig Ziggy's famous yogurt blends regular yogurt that is 3% fat with its no fat yogurt to obtain low fat yogurt that's 1% fat. How many pounds of no fat yogurt should be mixed to obtain 60 lbs of low fat yogurt? Wow, there's a lot of words there. So sometimes a little trick is to read the very ending and figure out what it's asking for. It wants to know how many pounds of no fat yogurt should be mixed to obtain 60 lbs. So I know that one of my unknowns is going to be amount of no fat yogurt. Now, it's usually two variable equations, so let's take this a little slower. So Ziggy's famous yogurt blends regular yogurt that's 3% fat. So if I have 3% fat, we need to know how much of the regular yogurt we're also getting because what's going to happen is our amount of no fat and our amount of regular is going to come together to make 60 per 60 lbs of the low fat. So no fat plus regular equals the thing in the middle. So when we look at the percentage, we know that regular yogurt is 3%, but how much percentage is no fat yogurt or how much fat is a no fat 0? So we're going to have percent, 0% times the amount plus 3% point, 03 times the amount equaling the percent times the amount of the mixture. So percent times amount, percent times amount, percent times amount. We do that because we weren't given a total like in the interest problem a moment ago, in a moment, the previous problem, they gave us the total interest. Now they're telling us how much of each piece 1% and 60 lbs. So percent amount, percent amount, percent amount. A clue or a trick is to think about this mixtures percent has to be between the other two percentages. So I've got to have a low percent and a high percent to get a mixture. And that percentage of the mixture has to be between. I'm going to go ahead and let you finish steps 3-4 and five. That was step one and that was Step 2. So solve for X, solve for Y, check it, and then answer what's asked for. And this next one, in a chemistry class, 12 liters of a twelve person alcohol solution must be mixed with a 20% solution to get a 14% solution. How many liters of the 20% solution are needed? So in a chemistry class, we're going to have 12 liters of 12% and we're mixing it with a 20%. So let X equal amount of 20% solution. So we've got 12 liters that we're mixing with 20% solution to get a 14% solution. So let's let Y equal the amount of 14% solution. So the 12 liters mixed with 20% solution quantity of liter equal the total liters. Liters plus liters equals liters. Now the percentage percent 12% times the amount of 12 liters plus the percent of 20% times the amount of X liters is going to equal the percent of 14 times the amount of Y. So percent times amount plus percent times amount equal percent times amount. Step one. Step 2. For this one, I might do substitution because we already had AY equal instead of elimination. Just putting the 12 + X in down here for the Y solving it. So you finish steps 3-4 and five. This last example, a boat travels 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 rate times time equal distance. So we have downstream and we have upstream. And the problem says find the speed of the boat in Stillwater and the speed of the current. Well, if we're going downstream, we have the speed of the boat and then the current is making us go faster. So let X equals speed of boat and Y be speed of current. If we're going upstream, that current is slowing us down or making us go slower. So it would be the speed of the boat minus the current. The time is 7 hours. So we're going to have the rate times the time for going downstream, the rate times the time going upstream. So this is distance down and this is distance up. What can we say about these two distances? These two distances are actually really the same. So we're going to have X + y * 7 equaling 168 / 2, or we could think of that just as 84, and then we're going to get X -, y * 42 equaling 84. So the first thing was to define our variables, which we did up here, and the second was to actually set up our equations. Now going to give a helpful hint before I let you finish this one up. See how do we have X + y * 7? I could divide each side by 7 and get X + y equaling 12. This X -, y * 42 equal 84. I could divide each side by 42 and get X -, y equaling 2. And finishing that up makes it a little easier than multiplying it out and getting bigger numbers. So finish up this worksheet. Thank you and have a wonderful day.