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Zorluk: OrtaSystems of Linear Inequalities in Two Variables

A school club is planning a fundraiser by selling two types of customized keychains: acrylic keychains and metal keychains. Acrylic keychains cost 3tomakeandsellfor3 to make and sell for 5, while metal keychains cost 5tomakeandsellfor5 to make and sell for 8. The club has a budget of at most 150toproducethekeychains,andtheywanttomakeatleast40keychainsintotal.Let150 to produce the keychains, and they want to make at least 40 keychains in total. Let x representthenumberofacrylickeychainsand represent the number of acrylic keychains and y$ represent the number of metal keychains. Which of the following systems of inequalities represents this situation?

  1. 3x+5y≤1503x + 5y \leq 150 and x+y≥40x + y \geq 40Cevap
  2. B
    3x+5y≥1503x + 5y \geq 150 and x+y≤40x + y \leq 40
  3. C
    5x+8y≤1505x + 8y \leq 150 and x+y≥40x + y \geq 40
  4. D
    5x+3y≤1505x + 3y \leq 150 and x+y≥40x + y \geq 40

Cevap

3x+5y≤1503x + 5y \leq 150 and x+y≥40x + y \geq 40
The correct system of inequalities must reflect the constraints on the budget and the total number of keychains. The budget is at most 150,whichmeansthetotalproductioncostmustbelessthanorequalto150.Sinceeachacrylickeychaincosts150, which means the total production cost must be less than or equal to 150. Since each acrylic keychain costs 3 to make and each metal keychain costs 5,thecostconstraintisrepresentedby5, the cost constraint is represented by 3x + 5y \leq 150 .Theclubalsowantstomakeatleast40keychains,whichmeansthetotalcountmustbegreaterthanorequalto40.Thisconstraintisrepresentedby. The club also wants to make at least 40 keychains, which means the total count must be greater than or equal to 40. This constraint is represented by x + y \geq 40 .Together,theseformthesystem. Together, these form the system 3x + 5y \leq 150 and and x + y \geq 40$.

Adım Adım Çözüm

1
Identify the variables and their meanings from the problem statement.
xx represents the number of acrylic keychains, and yy represents the number of metal keychains.
Understanding the variable definitions is necessary to construct the constraints correctly.
2
Set up the budget inequality using the production costs.
3x+5y≤1503x + 5y \leq 150
Each acrylic keychain costs 3tomake,eachmetalkeychaincosts3 to make, each metal keychain costs 5, and the total cost must be at most (less than or equal to) $150.
3
Set up the quantity inequality.
x+y≥40x + y \geq 40
The club wants to make a total of at least (greater than or equal to) 40 keychains.

Anahtar Kavram

Systems of linear inequalities in context
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Konular

Systems of Linear Inequalities in Two Variables

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