https://file.aiquickdraw.com/mathbot/file/1725419563999xbrpj536.jpg

The image contains a mathematical problem written in Indonesian. The problem is asking to find the maximum value of a function subject to certain constraints. Here is the translation and interpretation:

**Problem:**
1. \( x + y \leq 6 \)
2. \( x + z \leq 2 \)

Find the maximum value of the expression \( 2 + 5x + 2y \).

### Solution:

This is a linear programming problem. Let's outline the steps to solve it:

1. **Constraints:**
   - \( x + y \leq 6 \)
   - \( x + z \leq 2 \)
   - \( x, y, z \geq 0 \) (assuming non-negativity constraints)

2. **Objective Function:**
   - Maximize \( 2 + 5x + 2y \).

3. **Method:**
   - We need to plot the constraints on the \(xy\)-plane (ignoring \(z\) since the objective function does not include \(z\)).
   - The intersection points of the lines formed by the constraints will be evaluated in the objective function to determine which gives the maximum value.

### Steps to Calculate:

1. **Graph the Constraints:**
   - \( x + y = 6 \) (Line 1)
   - \( x = 2 \) (since \( z \) is not in the objective function, the constraint \( x + z \leq 2 \) simplifies to \( x \leq 2 \)).

2. **Find Intersection Points:**
   - The lines intersect at points \( (0, 6) \) and \( (2, 4) \) (from \( x + y = 6 \) and \( x = 2 \)).

3. **Evaluate the Objective Function:**
   - At \( (0, 6) \): \( 2 + 5(0) + 2(6) = 2 + 0 + 12 = 14 \).
   - At \( (2, 4) \): \( 2 + 5(2) + 2(4) = 2 + 10 + 8 = 20 \).

The maximum value is **20** at the point \( (2, 4) \).

### Conclusion:
The maximum value of \( 2 + 5x + 2y \) given the constraints is **20**.

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Do you want more details or have any questions?

Here are some related questions to consider:
1. What is linear programming, and how does it work?
2. Can the problem be solved using other methods like the Simplex algorithm?
3. How would the solution change if additional constraints were added?
4. What if the objective function was different, e.g., \( 3x + 4y + z \)?
5. How do changes in the coefficients of the objective function affect the maximum value?

**Tip:** Always check the feasibility of intersection points by ensuring they satisfy all constraints before evaluating the objective function.

Tentukan nilai maksimum dari 2 + 5x + 2y dengan syarat x + y ≤ 6 dan x + z ≤ 2.

Discover how to find the maximum value of the expression 2 + 5x + 2y given the constraints x + y ≤ 6 and x + z ≤ 2 using linear programming techniques. Ideal for students in Grades 10-12.

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