For each problem, write the constraints and graph the...

For each problem, write the constraints and graph the solution set to the system of constraints. See Example 2 .
Let $P(x, y)=6 x+8 y, R(x, y)=11 x+20 y,$ and $C(x, y)=$
$5 x+12 y .$ Evaluate each expression. See Example 3
$$P(1,5)$$

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Key Concepts

Linear Function Evaluation

Evaluating a linear function involves substituting specific values for its variables into the function's expression. This process simplifies the function into a numerical result, providing important insights such as checking objective function values at specific points in optimization problems.

System of Constraints

A system of constraints, often expressed as a set of linear inequalities, defines the feasible region within which solutions must lie. These constraints represent conditions such as resource limitations, non-negativity, or other requirements that the variables must satisfy in practical problems like optimization or linear programming.

Graphing the Solution Set

Graphing the solution set involves plotting each constraint on a coordinate plane and then identifying the region where all constraints overlap. This graphical method helps visualize the feasible region which is typically a polygon (or sometimes unbounded) formed by the intersection of the half-planes defined by the inequalities.

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