Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Story Problems in three variables. First thing we're going to do is we're going to define all of our variables, XY and Z. Then we're going to set up our equations. The sum of the three numbers is five. So we know that the sum which is addition X + y + C is five. The first number minus the 2nd plus the third is one, and the first minus the third is three more than the second. Now the first thing that we should think about is we want all the variables on one side. So we're going to subtract that Y over so that we get all the variables on one side for all three equations. Now we want to think about which variable do we want to eliminate. We're going to look for some that have a positive coefficient, some that have a negative. So let's look at maybe eliminating the YS. If we eliminate the YS, we have equation 1-2 and three. If we combine one and two exactly the way they are, we will get the YS to cancel and we'll get a new equation that only has XS and ZS. SO2X plus two Z = 6. Now, if we wanted to, we could actually divide that all through by two and get an equivalent equation, and we'll call that equation 4. Going back to the original three, I need to use the third equation because I haven't used it yet with one of the other two. So let's combine it with two, one, and two. If I add that, the YS will go away again. It's important that we always stick with eliminating the same variable. So we'd get 2X equaling 80. the Y and the Z canceled this time. So if two X = 8, we know that X equal 4. Now, if we know X, we have this equation #4 that had two variables in it, X&Z. So we're going to say 4 + Z equal 3, and we're going to solve for Z and get -1 Now we know two of our three unknowns, so I can stick it back into any of the original equations and find the third unknown. So 4 + y + -1 equal 5, or 3 + y equal 5 Y equal 2. Now, in a story problem, we always want to check, and we want to make sure it works in all three equations. So does 4 + 2 - 1 equal five? Well, that's 5 equal 5. So yes, that one checks. How about does 4 - 2 + -1 equal one? That one also checks and finally does 4 -. A negative 1 equal 2 + 3. Well, minus a negative is the same thing as a plus, so 5 equal 5. The last step in a story problem is to answer what's asked for in words. So the three numbers are 4, two, and -1. The New York Knicks recently scored a total of 92 points on a combination of two point field goals, 3 point field goals, and one point foul shots. All together, the Knicks made 50 baskets and 19 more two pointers than foul shots. How many shots of each kind were made? Well, the first thing that we're going to do is define our variables. The next thing is that we need to think about how many points. Well, each two point field gold had two points associated with it. So let's say they made five two point field golds. There'd be a total of 10 points. So here we're going to have 2 * X, or the number of points times the actual number of field golds made. Then we're going to have three points for each of the Y field goals made and one point for each of the foul shots made. The total number of points should be 92. We also know that the total number of baskets or X + y + C is 50. And then nineteen more two pointers than foul shots, 19 more 2 pointers. So the two pointers was more. So we need to take the smaller 1 and add something to it to get to the bigger one. Now we want to make sure we get all of our variables on one side. SO2X plus three y + C equal 90, two X + y + C equal 50, and X -, C equal 19. Now in this equation or these systems, we actually want to eliminate the YS. And the reason is that that third equation already has AY missing. So if I just combine equations one and two, I'll get a new equation that also has AY missing and I can immediately combine it with equation 3. So I'm going to multiply this equation 2 by a -3 in order to make my YS go away. When I add this, I'm going to get negative X -, 2 Z equaling -58. Now I can combine that immediately with that X -, C equal 19 because negative X -, 2 Z equal -58. When I add those, the XS go away and we get -3 Z equal -39. Divide each by -3 and we get Z equal 13. Once I know one variable, I'm going to look back and find an equation that only had two variables so that I can substitute in the one that I know in order to solve for the other. Add 13 to each side and we're going to get X as 32. Now if I know X and ZI can go back to the original, any of them that have AY in it and solve for the Y. So 32 + y + 13 equal 50. That's 45 + y equal 50, so Y equal 5. Now we need to check it. We need to check it in all three equations. So to check does 2 * 32 + 3 * 5 + 13 give us 92? Well, this is 64 + 15 + 13 equal 92 and the answer's yes. Does 32 + 5 + 13 equal 50? And that answer is yes also. And does finally 32 equal 13 + 19 and yes. So the last thing we need to do is answer what it was asked and it says how many shots of each kind. So there were 32 two point field goals. There were five three-point field goals and 13 one point Val shots in a triangle AB and C The measure of angle B is twice the measure of angle A, so we have one equation that says B is twice A is usually means equals. In a math problem, the measure of angle C is 80 more than A, and the third equation is understood that all three angles of a triangle add up to 180. Now this problem, because we have B&C already equaling something in terms of our same third variable. So B isn't something in terms of A and C isn't something in terms of A. We're going to use what's called substitution here. We're going to leave A and instead of B we're going to put in what B equaled, which was 2A, and instead of C we're going to put in what C equal, which was A + 80. So now if we add this up, we get four, as if we take the +80 to the other side, we get 100. Divide by 4 and we get a equal 25. Once we find A, we can find B because B was just twice whatever A was, so twice 25, which would equal 50. And we can find C because C was just 80 more than angle A, so 105. So now we need to do a check. If we check it, does 50 equal 2 * 25? Yes. Does 105 equal 25 + 80? Yes again. And this 25 + 50 + 105 equal 180. And the answer's yes. So we're going to answer what was asked for. The angle A is 25°, angle B is 50° and angle C is 105°. In a recent year, companies spent a total of 84.8 billion on television, newspaper and radio ads. So if we let X Equal newspaper, Y Equal Television and Z Equal radio, we know that a total of 84.8 was spent. The total amount spent on television and of radio, so y + C is 2.6 more than what was spent on the newspaper, so X + 2.6. The amount spent on the newspaper ads X was 5.1 more than on television, so y + 5.1. The first thing we want to do is we want to get all the variables on one side. So X + y + Z equal 84.8, negative X + y + Z equal 2.6 and X -, y equal 5.1. When we're looking to determine which variable to eliminate, we want to use the Z's because the Z was already gone in that third equation. So if we combine equations one and two, we're going to get our Z to go away by multiplying the second equation through by a -1. That way we have additive inverses, so X -, y -, C equal -2.6. When we add these, the YS and the Z's are going to cancel, so we're going to get 2X equaling 82.2. If we divide by two, we get 41.1. Now we know our X, so we're going to substitute in to an equation that has two unknowns, one of them being the X. So we knew that X -, y equal 5.1, so 41.1 - 5.1 would equal a positive Y. So we're going to get 36 equaling Y. Now if we know X&Y, we're going to put it into one of the original equations that had AZ in it. So 41.1 + 36 + Z equal 84.8. So this is going to give us 77.1 + Z equaling 84.8. And if we subtract 77.1, we're going to get 7.7. The next thing we need to do is we need to check this. So for our check, does 41.1 plus 36 + 7.7 equal 84.8? And the answer is yes, it does. Then we're going to do 36 + 7.7 and does that equal 41.1 + 2.6. Well, over here we're going to get 43.7 and that does indeed check also. Lastly, does 41.1 equal 36 + 5.1? The answer there is yes. So our last step is to answer what was asked for and it says how much was spent on each form of advertising. So 41.1 billion spent on newspaper advertising, 36 billion spent on television advertising, and finally 5.1 billion spent on radio advertising. Thank you and have a wonderful day.