https://file.aiquickdraw.com/mathbot/file/1727323796716lchwyeq3.jpg

The image shows a system of linear inequalities with a graph plotting the feasible region for the following constraints:

1. \( 5x + 4y \leq 20 \)
2. \( 4x + 6y \leq 24 \)
3. \( x \geq 0 \) (non-negative \(x\)-axis)
4. \( y \geq 0 \) (non-negative \(y\)-axis)

Let's break down the system:

### Step 1: Analyze the Boundary Lines

1. **For \(5x + 4y = 20\)**:
   - If \(x = 0\), \(y = \frac{20}{4} = 5\), so one intercept is \( (0, 5) \).
   - If \(y = 0\), \(x = \frac{20}{5} = 4\), so the other intercept is \( (4, 0) \).
   - This line passes through points \( (0, 5) \) and \( (4, 0) \).

2. **For \(4x + 6y = 24\)**:
   - If \(x = 0\), \(y = \frac{24}{6} = 4\), so one intercept is \( (0, 4) \).
   - If \(y = 0\), \(x = \frac{24}{4} = 6\), so the other intercept is \( (6, 0) \).
   - This line passes through points \( (0, 4) \) and \( (6, 0) \).

### Step 2: Determine the Feasible Region

The graph represents a bounded region in the first quadrant due to the constraints \( x \geq 0 \) and \( y \geq 0 \).

- The boundary lines \( 5x + 4y = 20 \) and \( 4x + 6y = 24 \) intersect within the region.
- The intersection of these lines is calculated by solving the system of equations. Solving the two equations will help to find this point of intersection.

### Step 3: Solve the System of Equations for Intersection

We solve:
1. \( 5x + 4y = 20 \)
2. \( 4x + 6y = 24 \)

Let's solve it step by step.

The intersection point of the lines \( 5x + 4y = 20 \) and \( 4x + 6y = 24 \) is \( \left( \frac{12}{7}, \frac{20}{7} \right) \), approximately \( (1.71, 2.86) \).

### Step 4: Identify the Feasible Region

The feasible region is bounded by:
- The \( x \)-axis (\( y = 0 \)),
- The \( y \)-axis (\( x = 0 \)),
- The line \( 5x + 4y = 20 \),
- The line \( 4x + 6y = 24 \).

The solution set (or the feasible region) is the area where all inequalities hold true, and it's likely a quadrilateral with vertices:
- \( (0, 0) \) (origin),
- \( (0, 4) \) (from \( 4x + 6y = 24 \)),
- \( \left( \frac{12}{7}, \frac{20}{7} \right) \) (intersection),
- \( (4, 0) \) (from \( 5x + 4y = 20 \)).

Would you like to further analyze the vertices of the feasible region or check any specific aspects?

Here are five related questions to explore this further:
1. What is the area of the feasible region formed by the constraints?
2. How do we graphically verify the solution set from the given constraints?
3. How would the region change if one of the inequalities was strict (e.g., \( < \) instead of \( \leq \))?
4. What are the effects of shifting one of the lines (e.g., changing \( 5x + 4y = 20 \) to \( 5x + 4y = 15 \)) on the solution set?
5. How can we optimize a linear function over the feasible region?

**Tip:** When solving systems of inequalities, always check the boundary conditions carefully to ensure all points within the feasible region satisfy the inequalities.

Himpunan Penyelesaian dari: 5x + 4y ≤ 20 ; 4x + 6y ≤ 24 ; x ≥ 0 ; y ≥ 0 adalah daerah yang dibatasi oleh dua garis dan sumbu x serta y.

This page explains how to solve a system of linear inequalities: 5x + 4y ≤ 20 and 4x + 6y ≤ 24. The solution set is graphically represented in the first quadrant, showing the feasible region where all conditions are satisfied. Suitable for Grades 10-12 students.

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