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. One package of pasta is the right amount to go with one jar of pasta sauce. What is the best numbers of packages of pasta and jars of pasta sauce to buy for the food pantry? How many packages of pasta? How many jars of pasta sauce? Explain your reasoning.

Answer:Solutions: (2,10), (4,5)Not solutions: (1,12), (6,10), (12,8), (18,6)Step-by-step 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[tex]x+3y\le 36.[/tex]2. Tyler can carry up to 20 pounds of food in his backpack, then[tex]x+1.5y\le 20.[/tex]You get the following system of inequalities:[tex]\left\{\begin{array}{l}x+3y\le 36\\ x+1.5y\le 20\end{array}\right.[/tex]Now substitute the coordinates of each point:(1,12): [tex]\left\{\begin{array}{l}1+3\cdot 12=37> 36\\ 1+1.5\cdot 12=19\le 20\end{array}\right.[/tex]False, because first inequality doesn't hold.(2,10): [tex]\left\{\begin{array}{l}2+3\cdot 10=32\le 36\\ 2+1.5\cdot 10=17\le 20\end{array}\right.[/tex]True, both inequalities hold.(4,5): [tex]\left\{\begin{array}{l}4+3\cdot 5=19\le 36\\ 4+1.5\cdot 5=11.5\le 20\end{array}\right.[/tex]True, both inequalities hold.(6,10): [tex]\left\{\begin{array}{l}6+3\cdot 10=36\le 36\\ 6+1.5\cdot 10=21> 20\end{array}\right.[/tex]False, because secondt inequality doesn't hold.(12,8): [tex]\left\{\begin{array}{l}12+3\cdot 8=36\le 36\\ 12+1.5\cdot 8=24> 20\end{array}\right.[/tex]False, because second inequality doesn't hold.(18,6): [tex]\left\{\begin{array}{l}18+3\cdot 6=36\le 36\\ 18+1.5\cdot 6=27> 20\end{array}\right.[/tex]False, because second inequality doesn't hold.