Problem 29 - Algebra for College Students

Question

Maximize the function \(f(x, y)=3 x+7 y\) in the region determined by the following constraints: 63 $$ \begin{aligned} 3 x+2 y & \leq 18 \\ 3 x+4 y & \geq 12 \\ x & \geq 0 \\ y & \geq 0 \end{aligned} $$

Step-by-Step Solution

Answer

Maximum value is 21 at the point (0,3).

Step 1: Identify the Constraints

The constraints given are: \(3x + 2y \leq 18\), \(3x + 4y \geq 12\), \(x \geq 0\), and \(y \geq 0\). These constraints define a region in the Cartesian plane within which we need to find the maximum value of the objective function \(f(x, y) = 3x + 7y\).

Step 2: Graph the Inequalities

Draw the lines \(3x + 2y = 18\) and \(3x + 4y = 12\) on a graph. These lines divide the plane into regions. Identify the feasible region by checking which side of the line each inequality satisfies and ensuring it is within the bounds \(x \geq 0\) and \(y \geq 0\).

Step 3: Determine the Feasible Region

After graphing the lines, the feasible region is the area where all constraints overlap. Calculate the intersection points of the lines to find the vertices of this region. For this problem, solve the system of equations \(3x + 2y = 18\) and \(3x + 4y = 12\) to find intersection points.

Step 4: Calculate Intersection Points

Set the equations equal to solve for intersection points: Solve \(3x + 2y = 18\) and \(3x + 4y = 12\) simultaneously. Subtract the second equation from the first to eliminate \(x\): \(-2y = 6\), so \(y = -3\). This solution is not feasible, so check alternative intersections with the \(x\) or \(y\)-axes within feasible limits.

Step 5: Evaluate the Objective Function at Vertices

Check the points of intersection found in the previous steps, ensuring they fall within the feasible region. These include intersections with the axes as well: \((0,0)\), intersections \((6,0)\) from \(3x + 2y = 18\) with \(y = 0\), and \((0,3)\) from \(3x + 4y = 12\) with \(x = 0\). Evaluate \(f(x, y) = 3x + 7y\) at each vertex.

Step 6: Calculate Maximum Value

The vertices to evaluate are \((6,0)\), \((0,3)\), and other feasible intersection points if any. Calculate: \(f(6,0) = 3(6)+7(0) = 18\) and \(f(0,3) = 3(0)+7(3) = 21\). Determine which of these provides the maximum value.

Key Concepts

Objective Function

In linear programming, the objective function is a critical component.It represents the quantity that requires optimization, often maximization or minimization.For this problem, the objective function is given by \( f(x, y) = 3x + 7y \).This equation needs to be maximized within a specific region determined by the constraints.
The coefficients 3 and 7 in the function indicate how much each unit of \( x \) and \( y \) contributes to the total value.The task is to find the combination of \( x \) and \( y \) which results in the highest possible value for \( f(x, y) \).This involves checking various combinations of \( x, y \) within the defined feasible region.

Feasible Region

The feasible region in any linear programming problem is diagrammed in a coordinate system.It represents all possible solutions that satisfy the given constraints.For this exercise, several inequalities intersect to form this region:

• \( 3x + 2y \leq 18 \)
• \( 3x + 4y \geq 12 \)
• \( x \geq 0 \) and \( y \geq 0 \)

The feasible region is confined to the area where all these inequalities overlap. It often appears as a polygon on the graph.In this scenario, solving the equations will help to define this polygon and identify its vertices.These vertices are potential candidates where the objective function may achieve a maximum.

Constraints

Constraints are the rules or limits within a linear programming problem that a solution must satisfy.They often take the form of linear inequalities.In our example, the constraints are:

• \( 3x + 2y \leq 18 \)
• \( 3x + 4y \geq 12 \)
• Non-negativity constraints, \( x \geq 0 \) and \( y \geq 0 \)

These constraints restrict the values \( x \) and \( y \) can take.The aim is to find the optimal point where all these constraints are satisfied simultaneously while maximizing the objective function.Constraints define the structure of the feasible region and thus help in identifying the optimal solution.

Intersection Points

Intersection points occur where the boundary lines of the constraints meet in a linear programming graph.These points are vital as they serve as the vertices of the feasible region polygon.By solving each pair of constraint equations together, you can identify these intersection points.
For example, the intersection of \( 3x + 2y = 18 \) and \( 3x + 4y = 12 \) does not provide a feasible solution here, as it results in \( y = -3 \).Instead, find each line's intersection with the axes, like \( (6,0) \) and \( (0,3) \).The reason why these are significant is that they might yield the maximum value when placed into the objective function.