Diketahui x dan y merupakan penyelesaian dari sistem pertidaksamaan x + y <= 4; 3x + y <= 6...

Introduction

In this tutorial, we will solve a system of inequalities involving two variables, x and y. The goal is to determine the maximum value of the objective function z = 2x + 3y, given the constraints of the inequalities. This is a common problem in algebra, particularly in the context of linear programming.

Step 1: Understand the System of Inequalities

First, we need to identify the inequalities that form our system:

1. x + y ≤ 4
2. 3x + y ≤ 6
3. x ≥ 0
4. y ≥ 0

Practical Advice

• Graphing: Visualizing the inequalities on a coordinate plane can help understand the feasible region where the solutions lie.
• Boundary Lines: Turn the inequalities into equations to find the boundary lines (e.g., x + y = 4 and 3x + y = 6).

Step 2: Graph the Inequalities

Next, we will graph each inequality to find the feasible region.

1. Graph x + y = 4:

   • Find intercepts:
     • When x = 0, y = 4 (point (0, 4))
     • When y = 0, x = 4 (point (4, 0))
   • Draw the line and shade below it.

2. Graph 3x + y = 6:

   • Find intercepts:
     • When x = 0, y = 6 (point (0, 6))
     • When y = 0, 3x = 6 → x = 2 (point (2, 0))
   • Draw the line and shade below it.

3. Include x = 0 and y = 0:

   • This means we only consider the first quadrant of the graph where both x and y are non-negative.

Practical Advice

• Use graph paper or a graphing tool for accuracy.
• Make sure to clearly mark the feasible region where all shaded areas intersect.

Step 3: Identify Vertices of the Feasible Region

Locate the vertices of the feasible region formed by the intersection of the lines and the axes.

1. Find Intersection of Lines:

   • Solve the equations:
     • x + y = 4
     • 3x + y = 6
   • By substitution or elimination, the intersection point is (2, 2).

2. Vertices to Check:

   • (0, 0)
   • (0, 4)
   • (2, 2)
   • (4, 0)

Practical Advice

• Make sure to double-check the calculations to confirm the intersection points.

Step 4: Evaluate the Objective Function

Now, evaluate the objective function z = 2x + 3y at each vertex.

1. For (0, 0):
   • z = 2(0) + 3(0) = 0
2. For (0, 4):
   • z = 2(0) + 3(4) = 12
3. For (2, 2):
   • z = 2(2) + 3(2) = 10
4. For (4, 0):
   • z = 2(4) + 3(0) = 8

Practical Advice

• Keep calculations organized to avoid confusion.

Step 5: Determine the Maximum Value

Compare the values obtained from the objective function evaluations. The maximum value is the highest result from the evaluations.

Conclusion

The maximum value of the objective function z = 2x + 3y, given the constraints, is 12 at the vertex (0, 4). This solution demonstrates how to graph inequalities, identify feasible regions, and evaluate an objective function in optimization problems.

Next steps could involve practicing with different sets of inequalities or exploring more complex linear programming scenarios.