Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Word problems with three variables. Our first example, the sum of 3 numbers is five. The first number minus the second plus the third is 1. The first number, the first minus the third is three more than the second. Find the numbers. So our first step, we're going to have five steps. Our first step is to just define what our unknowns are. So let X be the first number, Y be the second, Z be the third. You could have used AB and C It doesn't matter what variables. Our second step is to come up with the actual equations. So the sum of the three numbers sum means addition X + y + C is five, then the first number minus the second plus the third is 1. So X -, y + C equal 1. The first minus the third is. The word is typically means equals, but not always. Three more than the 2nd, so y + 3. Three more than what the second was. The first thing I would recommend is to get all the XYS&Z's on one side. So in this third equation, I'm going to make it X -, y -, C equaling 3. So now we want to look for a variable to get rid of. Remember we want to find one coefficients and some that are positive and some that are negative. So maybe we'll do the Y's in this case. I've got some positive, I've got some negative. So if this was equation one and two and three, first thing to do is combine one and two. If I combine one and two, I can see the YS that cancel and that would give me a 2X plus two Z = 6. Now sometimes we can see that an equation can be simplified even further if I divide this by a two. So my new equation 4 is going to be X + Z equal 3. Going back, I'm going to combine one and three this time because I have a positive Y and a negative Y so I'm going to get 2X. And in this case the Z's are actually going to cancel also. So two X = 8, X equal 4. If X is 4, I can substitute that back into the equation 4. So 4 + Z equal 3 Z must be -1 and if I know X and ZI can find Y, so 4 + y + -1 equal 5 SY has to be two. So then our. That was all. Our third step. Our third step whole thing was to solve. Our 4th step was to check. So does 4 plus 2 + -1 equal 5? Yes, Does 4 - 2 + -1 equal 1? Yes. And does 4 - -1 equal 2 + 3? And the answer is yes. I try to recommend always to check in the original 3 equations, because if any of our moving things around or multiplying changes it, it might check in a further along equation, but not in the originals. The last thing is to answer what's asked for. So the three numbers are four, two, and -1. Looking at another example, the New York Knicks recently scored a total of 92 points on a combination of 2.3 point and one point foul shots. All together, the Knicks made 50 baskets. So the total of baskets is going to be X + Y + Z equal 50. Now the points would be two points for each X and three points for each Y and one point for each C for a total of 92 points, 19 more two pointers all together. The next made 50 baskets and 19 more two pointers than foul shots. So 19 more two pointers than foul shots. X was my two pointers and Z was my foul shots. I think this thing through as a balance. If I have more two pointers so this one's more right now, how do I make them balanced? Well, if this is already more, I would add the smaller amount to the smaller portion. So the smaller plus the 19 would equal the one that was more. So if I had 19 more two pointers, I have to add the 19 to the foul shots to get the bigger number. Or I could think of it as X -, 19 to equal Z. I'm going to let you go ahead and solve this #3 and triangle ABC. The measure of B is twice a, so B equal to A. The measure of C is 80 more than the measure of A in a triangle. The piece that you may not be aware of is that all three angles add up to 180°. Now this is one where elimination is not going to necessarily be the easiest. And the reason is if we look here, B&C are both in terms of A. So if I said A and instead of BI put in 2A and instead of CI put in A + 80 and then solve this, I'd get 4A equaling 100 or A is 25. Then I could see that B has to be 50 and C is going to be 80 + 25 or 105. Make sure to check it in all three equations and then answer what's asked for so you finish that one up. Example four. In a recent year, company spent a total of 884.8 billion on newspapers, televisions and radios. So X + Y + Z equal 84.8 the total amount spent on television and radio. So television and radio was only 2.6 more than the amount spent on the newspaper alone. So 2.6 is going to go with the smaller amount to make them equal. And then the amount spent on newspaper ads. Newspaper ads was 5.1 more than the amount spent on television. This one I would probably encourage you to solve by elimination by taking all the variables to the same side. So negative X + y + C equal 2.6 and then X -, y equal 5.1. To solve this one, I would encourage I would continue on by combining one and two to get rid of the Z because the Z is already gone in that third equation. So if we combine one and two, let's say X + y + Z equal 84.8 and negative X which would turn into positive X minus y -, C equal -2.6. If we finish solving that from there, we get 2X equaling 82.2, so X is 41.1. Now we could actually put that X straight into equation 3 to get Y substituted into either the other two to get C. Please finish this one up and then try #5 on your own. Thank you and have a wonderful day.