Beth is purchasing chocolate and vanilla ice cream cups for a birthday party. she needs a total of 24 ice cream cups and twice as many chocolate cups as vanilla cups. set up the two equations that can be used to find the number of each flavor of cups that beth needs to purchase. let the equation that represents the total number of ice cream cups be referred to as constraint 1. let constraint 2 refer to the equation that describes the ratio of the number of chocolate cups to vanilla cups. only constraint would be met if 9 vanilla cups and 18 chocolate cups were purchased. only constraint would be met if 6 vanilla cups and 18 chocolate cups were purchased.

Let x represent the total number of chocolate cups and y represent the total number of vanilla cups Beth buys.

She needs a total of 24 ice cream cups. So, the equation can be written as:



The number of chocolate cups she needs are twice as compared to vanilla, so we can write the equation as:



Case 1: 
Only constraint  2 would be met if 9 vanilla cups and 18 chocolate cups were purchased.
This is because the number of chocolate cups are double as compared to vanilla cups but the total number of cups in this case is not 24. So constraint 1 is not met. 

Case 2: 
Only constraint 1 would be met if 6 vanilla cups and 18 chocolate cups were purchased.
This is because the total number of cups in this case is 24 but the chocolate cups are three times as compared to the vanilla cups. So the constrain 2 is not met in this case.

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