Question

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

Accepted Answer

Step 1: Identify the constraints and the objective function. The objective function is given by $$z = 5x - y. $$The constraints are: $$3 \leq x \leq 7$$, $$y \geq 4$$, and $$-2x + 2y \leq 8. $$Step 2: Graph the inequalities on the coordinate plane. Start by drawing the vertical lines $$x = 3$$ and $$x = 7 to $$represent the bounds on $$x. $$Then, draw the horizontal line $$y = 4 to $$represent the lower bound on $$y. $$Finally, rewrite the inequality $$-2x + 2y \leq 8 as$$ $$y \leq x + 4$$ and graph the line $$y = x + 4. $$Shade the region below this line. Step 3: Determine the feasible region by finding the intersection of all shaded areas that satisfy all constraints simultaneously. This region will be bounded by the lines $$x=3$$, $$x=7$$, $$y=4$$, and $$y = x + 4. $$Step 4: Find the corner points (vertices) of the feasible region by solving the systems of equations formed by the intersection of the boundary lines. These points are where the maximum or minimum values of the objective function can occur. Step 5: Evaluate the objective function $$z = 5x - y at $$each corner point found in Step 4. Compare these values to determine which one gives the maximum value of $$z$$, and note the corresponding values of $$x$$ and $$y.$$

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Key Concepts

Graphing Systems of Linear Inequalities

Corner Points (Vertices) of the Feasible Region

Evaluating the Objective Function