# Monday, March 6

In class we are about to wrap up our unit on Systems of Equations and Inequalities. Students should be able to solve a system of equations by

• Making a graph
• Using the substitution method
• Using the elimination method

The unit also began by leading students to be able to solve systems of inequalities by graphing the inequalities and shading the graph to find the feasible region. This skill is also useful in solving maximization problems. In our problem set today students had to write a set of inequalities to find the optimal number of two sizes of cookies for a shop to make in order to maximize profits given three different sets of constraints.

Full size of this example here. In order to solve this kind of problem students need to

• Identify the appropriate variables to use in representing the constraints and profit — select variables which represent the quantities you are looking to balance (in this case small/large cookies)
• Represent each constraint using an inequality using the variables you defined above. For example, in this problem, since Jorge has 50 raisins, we notice that each large cookie uses 8 raisins, and each small: 3 raisins. So we can write $\dpi{300}\inline 3x + 8y \leq 50$
• Graph the constraints and identify the feasible region — where all points represent viable values of $\dpi{300}\inline x$ and $\dpi{300}\inline y$. Pairs which satisfy all constraints.
• Locate the vertices of the feasible region and compute the profit at each of these points. In a linear optimization problem, the maximum profit will be found either at one vertex of the feasible region or along an edge of the feasible region.

Assignment: First six problems in the handout from class.