Systems of inequalities word problems answer key

Video transcript

- [Voiceover] "Fleur wants to make tables and chairs. "Each chair or table is made with the same number "of wooden boards and nails. "She has a total of 150 wooden boards and 330 nails. "The following inequality represents the number "of tables, T, and chairs, C, "she can make with 150 wooden boards." So we can see the 150 wooden boards right over here and it looks like she uses 17 wooden boards for each table so the total amount of boards from the tables are to make the tables is 17 T. And it looks like she uses six wooden boards for each chair and that's why you take six per chair times the number of chairs. This is the total number of wooden boards she uses from the chairs. And you add them together. It can't be anymore than 150. She only has, I guess 150 wooden boards. "Additionally, the following inequality represents "the number of tables and chairs "she can make with 330 nails." So once again, she has the maximum of 330 nails. It looks like she needs 34 nails per table because the total nails from the tables is 34 nails per table times the number of tables. And then it looks like she needs 27 nails. It looks like she needs 27 nails per chair because the total nails that she uses in all of the chairs are 27 nails per chair times the number of chairs. "Does Fleur have enough boards "and nails to make three tables, "to make three, "to make three tables and nine chairs? "And nine chairs?" Let's see, let's look at the number of boards. To make three tables and nine chairs, she's going to use 17 times T. In this case, it's three. Shes' going to use 17 times three boards for the tables and then she is going to use, she is going to use six times the number of chairs. So six times nine chairs. Six boards per chair times nine chairs. So six times nine is the number of boards she's going to use on the chairs. So this is going to, this is the same thing as 17 times three is 51. 51 plus, six times nine is 54. So this is 54. And what is this? Is this less than or equal to 150? Is that less than or equal to 150? Let's see, 51 plus 54 is going to be 105. So that indeed is less than or equal to 150. So that checks out. So she has more than enough boards. So she has enough boards. Let me write that. Enough boards. Now let's see if she has enough nails. Let's see if she has, that came out weird, enough boards. Let's see if she has enough nails. So three tables are going to require 34 times three nails. So 34 nails per table times three tables. So 34 times three. That's how many nails for all of the tables. And then plus, if she is going to use 27 nails per chair times the nine chairs. 27 times nine. We need to figure out, is this less than or equal to 330? And so let's see. 34 times three, that is 100. No, that's 90 plus 12. So that's 102 and then 27 times nine. So let's see, that would be, was it 243? Yep, so plus 243. So it's 102 plus 243, is that less than or equal to 330? So let's see, this would be, this would end up adding up to 345. No, this is not less than or equal to 330. This is not the case. This is not true. Not true. So she does not have enough nails. So not enough, not enough nails. And so she has enough boards but not enough nails. So unless she goes and buys some nails, she's not going to be able to make these tables and chairs.

Video transcript

- [Instructor] We're told, "Luis is cooking meals for at least 20 people. He estimates that the cost of each vegetarian meal is $3, and the cost of each meal with meat is $4.50. His budget for the meals is no more than $100, and he wants to cook at least six of each type of meal. Which of the following systems of inequalities represents the conditions described if x is the number of vegetarian meals and y is the number of meals with meat Luis cooks?" So pause this video and have a go at this yourself before we work through it together. And I know this is a long question and these systems feel complicated, but trust me, if you do it step-by-step you'll actually find that it all falls into place. All right, now let's work through it together, and it's important to emphasize that they've already defined the two key variables for us. x is the number of vegetarian meals and y is the number of meals with meat. So let's look at each of the constraints they give us and each of these can set up a different inequality. So the first one is, they say, "Luis is cooking meals for at least 20 people." So that tells us that the total number of meals, which is going to be the number of vegetarian meals, that's x, plus the number of meat meals, that has to be at least equal to 20. So that has to be greater than or equal to 20. That's what that first sentence tells us. And if I wasn't doing this as a multiple choice, I would just keep adding more and more constraints here, but they give us some choices. And so we can see that x plus y is greater than or equal to 20, that's in choice A. It's actually not in choice B. So we can already rule out choice B. They have less than or equal to 20 here. Same thing for choice C. So we can rule that out. And then choice D does have that. So we are still in the running. The next constraint they tell us, "He estimates that the cost of each vegetarian meal is $3, and the cost of each meal with meat is 4.50. His budget for the meals is no more than $100." So how much is he gonna spend in total? Well, on the vegetarian meals, he's going to spend the number of vegetarian meals times $3 per meal. So that's how much he's going to spend on vegetarian meals. And what about meat meals? Well, it's going to be y meals times 4.50 per meal. So it's 4.5 times y. The amount that he's spending on vegetarian meals, the amount that he's spending on non-vegetarian meals, that's the total he's spending on meals. And they say it is no more than $100. So this has to be less than or equal to 100. And so let's see, we actually over here, we have 3x plus 4.5y is greater than or equal to 100. So we can rule choice A out as well and just by deductive reasoning, we see choice D does have that in there. But this must be the answer, but let's keep going to make sure that these other constraints work. We are also told he wants to cook at least six type of each meal. So that means that x, the number of vegetarian meals, has to be greater than or equal to six. And y, the number of non-vegetarian meals, also has to be greater than or equal to six. And we see both of these down here, so we can feel pretty good about choice D.