# Question Video: Forming a System of Linear Inequalities Mathematics • 9th Grade

David is going to the store to buy candles. Small candles cost \$3 and large candles cost \$5. He needs to buy at least 20 candles, and he cannot spend more than \$80. Write a system of linear inequalities that represents the situation, using 𝑥 to represent the number of small candles and 𝑦 to represent the number of large candles.

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### Video Transcript

David is going to the store to buy candles. Small candles cost three dollars and large candles cost five dollars. He needs to buy at least 20 candles, and he cannot spend more than 80 dollars. Write a system of linear inequalities that represents the situation, using 𝑥 to represent the number of small candles and 𝑦 to represent the number of large candles.

A system of inequalities can help us to solve various optimization problems. In this case, we usually begin by defining any relevant variables. However, we’ve been given a pair of variables 𝑥 and 𝑦, and we’re told that 𝑥 represents the number of small candles David will buy and 𝑦 represents the number of large candles.

With this in mind, let’s now identify the constraints we have. We’re told that he’s going to buy at least 20 candles. In other words, the total number of small candles and large candles he buys must exceed or be equal to 20. Well, since 𝑥 is the number of small candles and 𝑦 is the number of large candles, the total number of candles that he will buy will be given by 𝑥 plus 𝑦. We know this must be at least 20. So we can say that 𝑥 plus 𝑦 must be greater than 20.

We’re also told that he cannot spend more than 80 dollars. And we’re given the cost of a small candle as three dollars and a large candle as five dollars. So since 𝑥 is the number of small candles, we need to ask ourselves, what is the total amount of money that David will spend on these small candles? The total amount of money that David will spend on small candles alone will be the product of the price per one candle and the number of candles there are. In dollars then, that’s going to be equal to three 𝑥.

Let’s now consider the same problem and establish the total amount of money in dollars that David will spend on large candles. He buys 𝑦 candles at a price of five dollars per candle. So the total amount of money that David will spend on all of these large candles will be the product of these two values. In dollars, that’s five 𝑦. So the total amount of money that David will spend will be the sum of these. It’s three 𝑥 plus five 𝑦. We’re told he cannot spend more than 80 dollars, so the sum of these must not exceed 80. In other words, it must be less than or equal to 80. And so we have our second inequality: three 𝑥 plus five 𝑦 is less than or equal to 80.

So we now have a pair of inequalities. The first was identified by considering the total number of candles he would buy. The second of these was around the constraint involving the total amount of money or the price of each candle. In a first glance, it might feel like we’re finished, but there is a constraint we can put on the number of small candles and the number of large candles individually. We know he cannot buy a negative number of candles. And so in fact, both 𝑥 and 𝑦 must be greater than or equal to zero.

So here is the system of four linear inequalities that represents this situation. 𝑥 is greater than or equal to zero, 𝑦 is greater than or equal to zero, 𝑥 plus 𝑦 is greater than or equal to 20, and three 𝑥 plus five 𝑦 is less than or equal to 80.