Gardening - Salsa or Spaghetti Sauce

Mr. McAdams grows tomatoes and peppers in his garden each year Last year he harvested 84 pounds of tomatoes and 12 pounds of bell peppers. He wants to make salsa and spaghetti sauce. The two tables give the ingredients for a single batch of salsa and spaghetti sauce.

Salsa 16 pounds tomatoes 1 pound bell peppers 4 onions 10 jalapeño peppers 2 tablespoons of salt 1 bunch cilantro 1 tablespoon oregano 5 garlic cloves

Spaghetti Sauce 8 pounds tomatoes 2 pounds bell peppers 1 onion 2 tablespoons oregano 2 tablespoons basil 2 tablespoons thyme

How much?

1. If Mr. McAdams makes 2 batches of salsa, how many pounds of tomatoes and bell peppers will he need?
2. If Mr. McAdams makes 3 batches of spaghetti sauce, how many pounds of tomatoes and bell peppers will he need?
3. Use to represent the number of batches of salsa and to represent the number of batches of spaghetti sauce. Write an equation showing 80 pounds of tomatoes needed for batches of salsa and batches of spaghetti sauce.
4. Write an equation showing the number of pounds of bell peppers to make batches of salsa and batches of spaghetti sauce.
5. How many batches of salsa and spaghetti sauce will use all the peppers and tomatoes? (Hint: Use two linear equations and a 2x3 matrix).

Linear Inequalities

A linear inequality is a linear system that uses inequalities ( ≤, <, ≠, >, ≥), and does not use equality (=). We will use the canning example from the last page. When we used the linear system 18x + 8y = 80 and 1x + 2y = 14, we were able to find out the number of batches that used all the tomatoes and peppers. We will now change the equations to inequalities. Since we can use at most 80 pounds of tomatoes, the first equation becomes 18x + 8y ≤ 80. The second equation becomes 1x + 2y ≤ 14.

1. Graph the two inequalities. (Hint: put each equation in slope intercept form.)
   
2. Find the point (2,6). What does this point represent?
3. Could Mr. McAdams make 4 batches of spaghetti sauce and 2 batches of salsa with the available ingredients? How do you know this?
4. Could Mr. McAdams make 3 batches of spaghetti sauce and 6 batches of salsa with the available ingredients? How do you know this?
5. The point (3,3) included in the region. What does it mean that the point (3,3) is included in the region?
6. The point (2,7) is not included in the region. What does it mean that the point (2,7) is not included in the region?
7. If Mr. McAdams wants to make exactly 3 batches of salsa, how many batches of spaghetti sauce should he make to use up the most ingredients?