Problem 12 - College Algebra

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. Objective Function \(\quad z=2 x+4 y\) \(\begin{array}{ll}\text { Constraints } & x \geq 0, y \geq 0 \\ & x+3 y \geq 6 \\ & x+y \geq 3 \\ & x+y \leq 9\end{array}\)

Step-by-Step Solution

Answer

The maximum value of the objective function and the corresponding values of \(x\) and \(y\) can be found once we have plotted the feasible region, identified the corners, and evaluated the objective function at these corners. The particular \(x\) and \(y\) pair that yields the maximum value for \(z\) is the solution to the problem.

Step 1: Graphing the Constraints

Plot the system of inequalities on the same set of axes. The inequalities represent half-planes. The solution region is the common area for all constraints. Exclude the regions for \(x<0\) and \(y<0\) since we only consider positive values of \(x\) and \(y\). The feasible region should be bounded by the lines \(x+3y=6\), \(x+y=3\), and \(x+y=9\). The point where these lines intersect forms a polygon, which is the feasible region of the system.

Step 2: Corners of the Feasible Region

The corners of the feasible region are the solutions to the system of equations formed by every pair of constraints. Therefore, find the intersection points of lines \(x+3y=6\), \(x+y=3\), and \(x+y=9\), along with axes. For example, to find the intersection between \(x+3y=6\) and \(x+y=3\), put the two equations together and solve for \(x\) and \(y\). Do the same for the remaining pairs, giving us our corner points.

Step 3: Evaluate the Objective Function at the Corners

Substitute the coordinates of corners of the feasible region into the objective function \(z=2x+4y\) to get the value of \(z\) at each corner.

Step 4: Determine Maximum Value

Compare the values of \(z\) obtained in the previous step to locate the maximum. The maximum value of the function is the highest value of \(z\) and the coordinates \((x,y)\) giving the maximum \(z\) are the solution.

Key Concepts

Objective Function

The objective function in a linear programming problem is a mathematical expression that we aim to maximize or minimize. In our exercise, the objective function is given by the equation:

\( z = 2x + 4y \)
This function represents the goal we want to achieve, such as maximizing profits or minimizing costs, through the combinations of variables (\(x\) and \(y\) in this case). Finding the most beneficial values of \(x\) and \(y\) that also satisfy the given constraints is what the linear programming problem is all about.

System of Linear Inequalities

A system of linear inequalities is a set of constraints that limit the feasible solutions of a linear programming problem. These constraints are expressed as inequalities that must be satisfied simultaneously. In the given exercise, we have four such inequalities:

\(x \text{≥} 0\), \(y \text{≥} 0\), \(x+3y \text{≥} 6\), and \(x+y \text{≤} 9\).
Essentially, these inequalities define a range of possible solutions for \(x\) and \(y\), and are graphical representations that will ultimately form a feasible region on a coordinate plane.

Feasible Region

The feasible region is the graphical representation of all possible combinations of \(x\) and \(y\) that satisfy the system of linear inequalities. It is usually a polygonal shape on the coordinate plane where each vertex, or corner point, represents a potential solution to the linear programming problem. The exercise directs us to find the feasible region bounded by the inequalities \(x+3y = 6\), \(x+y = 3\), and \(x+y = 9\), along with the non-negativity constraints.

Graphing Inequalities

To graph inequalities, we first graph the corresponding linear equations as if the inequalities were equal signs. These lines divide the plane into two halves, one of which will satisfy the inequality. In the exercise, these lines determine the boundary of our feasible region. We also note that the non-negativity constraints confine our solutions to the first quadrant where both \(x\) and \(y\) are non-negative. To find the shaded feasible region, we identify the shared area that fulfills all inequalities. This task forms the basis of visualizing the limits within which our objective function can be optimized.

Optimization

Optimization in linear programming is the process of finding the best, or 'optimal', value for the objective function within the feasible region. After graphing our feasible region, we evaluate the objective function at each vertex. This is because, in linear programming, the optimal value will always occur at a vertex of the feasible region. In this exercise, it involves calculating the value of \(z = 2x + 4y\) for each corner point. The maximum (or minimum, depending on the problem) of these values corresponds to our optimal solution. The exercise asks us to identify the corner that yields the maximum value of the objective function and the specific values of \(x\) and \(y\) at that point.