Question

parakeets pet store never has more than a combined total of 20 cats and dogs and never more than 8 cats. given that x represents the number of cats at the pet store and y represents the number of dogs at the pet store, select all possible solutions. (8, 12) (18, 0) (12, 8) (0, 18) (9, 11) (6, 14) (7, 15) (14, 6)

Explanation:

Step1: Analyze the constraints

We have two constraints: \( x + y \leq 20 \) (combined total of cats and dogs is at most 20) and \( x \leq 8 \) (number of cats is at most 8), where \( x \) is the number of cats and \( y \) is the number of dogs, and \( x,y\geq0 \) (since the number of animals can't be negative).

Step2: Check each ordered pair \((x,y)\)

• For \((8, 12)\):
• Check \( x \leq 8 \): \( 8\leq8 \), which is true.
• Check \( x + y \leq 20 \): \( 8 + 12 = 20 \leq 20 \), which is true. So \((8, 12)\) is a solution.
• For \((18, 0)\):
• Check \( x \leq 8 \): \( 18\leq8 \), which is false. So \((18, 0)\) is not a solution.
• For \((12, 8)\):
• Check \( x \leq 8 \): \( 12\leq8 \), which is false. So \((12, 8)\) is not a solution.
• For \((0, 18)\):
• Check \( x \leq 8 \): \( 0\leq8 \), which is true.
• Check \( x + y \leq 20 \): \( 0+18 = 18\leq20 \), which is true. So \((0, 18)\) is a solution.
• For \((9, 11)\):
• Check \( x \leq 8 \): \( 9\leq8 \), which is false. So \((9, 11)\) is not a solution.
• For \((6, 14)\):
• Check \( x \leq 8 \): \( 6\leq8 \), which is true.
• Check \( x + y \leq 20 \): \( 6+14 = 20\leq20 \), which is true. So \((6, 14)\) is a solution.
• For \((7, 15)\):
• Check \( x \leq 8 \): \( 7\leq8 \), which is true.
• Check \( x + y \leq 20 \): \( 7 + 15=22>20 \), which is false. So \((7, 15)\) is not a solution.
• For \((14, 6)\):
• Check \( x \leq 8 \): \( 14\leq8 \), which is false. So \((14, 6)\) is not a solution.

Answer:

The possible solutions are \((8, 12)\), \((0, 18)\), and \((6, 14)\). So the selected options should be:

• \(\boldsymbol{(8, 12)}\)
• \(\boldsymbol{(0, 18)}\)
• \(\boldsymbol{(6, 14)}\)