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The objective function in linear programming is a mathematical expression that you want to optimize, typically maximize or minimize. In the given problem, the objective function is \( z = 3x + 2y \). This means we want to determine the values of \( x \) and \( y \) that will either maximize or minimize \( z \).

The coefficients of the variables \( x \) and \( y \) (3 and 2, respectively) are critical in understanding the impact of changing these variables on the objective. Here, if \( x \) increases by 1 unit, \( z \) increases by 3. Similarly, a 1 unit increase in \( y \) results in a 2 unit increase in \( z \).

In essence, the objective function acts as a guiding mathematical tool, directing us towards optimal decision-making within the constraints provided.

The feasible region in a linear programming problem is the set of all possible points that satisfy all given constraints simultaneously. This is the area within which any combination of \( x \) and \( y \) is considered valid, adhering to the problem constraints.

In this scenario, the constraints are:

• \( x \geq 0 \)
• \( y \geq 0 \)
• \( 5x + 2y \leq 20 \)
• \( 5x + y \geq 10 \)

These constraints shape our feasible region, which, according to the solution, lies within the first quadrant of the graph. It is bounded by lines that meet due to the inequalities, capturing a designated space known as the feasible region.

This region is crucial because it houses potential solutions to our problem, ensuring that only practical and realistic solutions are considered when optimizing the objective function.

Graphing inequalities involves plotting the constraints on a coordinate plane to visually represent the feasible region. First, we plot each constraint equation as if the inequalities were equalities:

• \( 5x + 2y = 20 \)
• \( 5x + y = 10 \)

Each equation is drawn as a straight line.

To represent the inequalities, we shade the side of the line that fulfills the inequality condition. For instance, with \( 5x + 2y \leq 20 \), the region below the line is shaded. Similarly, for \( 5x + y \geq 10 \), we shade above the line.

By considering \( x \geq 0 \) and \( y \geq 0 \), we restrict our focus to the first quadrant. The intersecting shaded regions from all inequalities highlight the feasible region, enabling us to visually locate all feasible solutions.

Vertices are the corner points of the feasible region found by the intersection of the constraint lines. They are significant because in linear programming problems, optimal solutions often occur at these points.

For instance, solving the constraints simultaneously helps in determining these intersections:

• The equation of \( 5x + 2y = 20 \) combines with \( 5x + y = 10 \) to find one such vertex.

From the solution, these vertices turn out to be:

• \((0, 0)\)
• \((2, 0)\)
• \((2, 5)\)
• \((0, 10)\)

Each vertex is assessed by calculating the objective function's value at that point. This allows us to easily compare which vertex optimizes the function to find either the maximum or minimum value.