Problem 9 - College Algebra Essentials

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 Constraints \(z=3 x-2 y\) \(\left\\{\begin{array}{l}1 \leq x \leq 5 \\ y \geq 2 \\ x-y \geq-3\end{array}\right.\)

Step-by-Step Solution

Answer

First plot the constraints on a graph to find the feasible region. Identify the corner points of this feasible region. Calculate the value of the objective function \(z=3x-2y\) at these corner points. The maximum value from these is the maximum of the function and the (x,y) pair for this maximum value is the solution.

Step 1: Graphing the Inequalities

Let's start with graphing the inequalities. This involves three inequalities \[1 \leq x \leq 5\], \[y \geq 2\] and \[x-y \geq -3\]. Plot these on a coordinate plane.

Step 2: Identify Cornwall Points

Upon successfully plotting all the inequalities, identify the corner points of the region. The graphical representation can offer clear visualization for identifying these points.

Step 3: Compute Objective function values

With these corner points, calculate the value of the objective function \(z=3x-2y\) at these point. Substitute the x and y values of each corner point in the objective function to get the z-value.

Step 4: Determine the Maximum Value

Compare the functions values that were computed in the previous step. The highest value among these will represent the maximum value of the objective function. Accordingly, the corresponding (x,y) pair where the maximum value is attained, is the solution.

Key Concepts

Objective Function

The concept of an objective function is central to linear programming. It is a mathematical expression you want to either maximize or minimize. In our case, the objective function given is \(z = 3x - 2y\). Here, \(x\) and \(y\) are variables which represent the decision points, and the coefficients (3 and -2) determine how much each variable contributes to the result or output \(z\). The ultimate goal is to find the values of \(x\) and \(y\) that either maximize or minimize this function. This makes the objective function a target we aim to achieve under specified constraints.

System of Linear Inequalities

A system of linear inequalities represents the constraints or limitations within which the variables must operate. In our example, we have three constraints: \(1 \leq x \leq 5\), \(y \geq 2\), and \(x - y \geq -3\). These inequalities form a system because they all must be satisfied simultaneously. Each inequality restricts the feasible region where the values of \(x\) and \(y\) can fall. It's important to understand that a system of inequalities defines a range of solutions, unlike equations which give precise solutions.

Graphing Inequalities

Graphing inequalities involves representing each constraint on a coordinate plane to identify the feasible region. This region is where all inequalities overlap or intersect. Here's how you can graph:

• For \(1 \leq x \leq 5\): Draw vertical lines at \(x = 1\) and \(x = 5\). Shade the region between them.
• For \(y \geq 2\): Draw a horizontal line at \(y = 2\) and shade above it.
• For \(x - y \geq -3\): Rearrange to \(y \leq x + 3\), draw the line \(y = x + 3\), and shade below it.

The intersection of these shaded areas is the feasible region. This is crucial for identifying where the solution points, known as corner points, may lie.

Corner Points

Corner points are the key aspects in finding the solution to a linear programming problem. These points occur at the intersection of the boundaries formed by the graphed inequalities. After graphing, you can visually identify these corners, which are the potential solution candidates. Each corner point will be a set of \((x, y)\) coordinates. In our exercise, you substitute these coordinates back into the objective function \(z = 3x - 2y\). These calculated \(z\) values will help determine the maximum (or minimum) objective function value. Evaluating the function at each corner point ensures you haven't missed the optimal solution.