Question

Tyler went to the supermarket to buy food for a food pantry. He has $36, and can carry up to 20 pounds of food in his backpack. Pasta costs $1 for a 1-pound package. Pasta sauce costs $3 for a 1.5 pound jar. Let x = the number of packages of pasta and y = the number of jars of pasta sauce. Identify each point as either a solution to the system or not a solution to the system of inequalities. (1, 12) (2, 10) (4, 5) (6, 10) (12, 8) (18, 6) .

Answer 1

Solutions: (2,10), (4,5)

Not solutions: (1,12), (6,10), (12,8), (18,6)

Explanation:

Let x be the number of packages of pasta and y be the number of jars of pasta sauce. If pasta costs $1 for a 1-pound package, then x packages of pasta cost $x and weigh x pounds. If pasta sauce costs $3 for a 1.5 pound jar, then y jars cost $3y and weigh 1.5y pounds.

1. Tyler has $36, then

`x+3y\le 36.`

2. Tyler can carry up to 20 pounds of food in his backpack, then

`x+1.5y\le 20.`

You get the following system of inequalities:

`\left\{\begin{array}{l}x+3y\le 36\\ x+1.5y\le 20\end{array}\right.`

Now substitute the coordinates of each point:

(1,12):

`\left\{\begin{array}{l}1+3\cdot 12=37> 36\\ 1+1.5\cdot 12=19\le 20\end{array}\right.`

False, because first inequality doesn't hold.

(2,10):

`\left\{\begin{array}{l}2+3\cdot 10=32\le 36\\ 2+1.5\cdot 10=17\le 20\end{array}\right.`

True, both inequalities hold.

(4,5):

`\left\{\begin{array}{l}4+3\cdot 5=19\le 36\\ 4+1.5\cdot 5=11.5\le 20\end{array}\right.`

True, both inequalities hold.

(6,10):

`\left\{\begin{array}{l}6+3\cdot 10=36\le 36\\ 6+1.5\cdot 10=21> 20\end{array}\right.`

False, because secondt inequality doesn't hold.

(12,8):

`\left\{\begin{array}{l}12+3\cdot 8=36\le 36\\ 12+1.5\cdot 8=24> 20\end{array}\right.`

False, because second inequality doesn't hold.

(18,6):

`\left\{\begin{array}{l}18+3\cdot 6=36\le 36\\ 18+1.5\cdot 6=27> 20\end{array}\right.`

False, because second inequality doesn't hold.