Solve the system of linear inequations: `x + 2y le 10, 2x + y le 8`.

Options

A. Lines meet at E(2,4)
B. Lines meet at E(-2,4)
C. Lines meet at E(2,-4)
D. Lines meet at E(-2,-4)

Answer

To solve the system of linear inequalities given by \( x + 2y \leq 10 \) and \( 2x + y \leq 8 \), we will follow these steps:

### Step 1: Write the inequalities
We have the following inequalities:
1. \( x + 2y \leq 10 \)
2. \( 2x + y \leq 8 \)

### Step 2: Convert inequalities to equations
To find the boundary lines, we convert the inequalities into equations:
1. \( x + 2y = 10 \)
2. \( 2x + y = 8 \)

### Step 3: Find the intercepts for each equation
**For the first equation \( x + 2y = 10 \):**
- To find the x-intercept, set \( y = 0 \):
\[ x + 2(0) = 10 \implies x = 10 \quad \text{(x-intercept: (10, 0))} \]
- To find the y-intercept, set \( x = 0 \):
\[ 0 + 2y = 10 \implies y = 5 \quad \text{(y-intercept: (0, 5))} \]

**For the second equation \( 2x + y = 8 \):**
- To find the x-intercept, set \( y = 0 \):
\[ 2x + 0 = 8 \implies x = 4 \quad \text{(x-intercept: (4, 0))} \]
- To find the y-intercept, set \( x = 0 \):
\[ 2(0) + y = 8 \implies y = 8 \quad \text{(y-intercept: (0, 8))} \]

### Step 4: Plot the lines on a graph
- Plot the points for both equations on a graph:
- For \( x + 2y = 10 \): Points (10, 0) and (0, 5)
- For \( 2x + y = 8 \): Points (4, 0) and (0, 8)

### Step 5: Shade the feasible region
- For the inequality \( x + 2y \leq 10 \), shade the area below the line.
- For the inequality \( 2x + y \leq 8 \), shade the area below the line.
- The feasible region is where the shaded areas overlap.

### Step 6: Identify the vertices of the feasible region
- The vertices of the feasible region can be found by determining the intersection points of the lines:
1. Solve the equations simultaneously:
\[ x + 2y = 10 \quad (1) \]
\[ 2x + y = 8 \quad (2) \]
From (2), express \( y \):
\[ y = 8 - 2x \]
Substitute into (1):
\[ x + 2(8 - 2x) = 10 \]
\[ x + 16 - 4x = 10 \]
\[ -3x = -6 \implies x = 2 \]
Substitute \( x = 2 \) back into (2):
\[ 2(2) + y = 8 \implies 4 + y = 8 \implies y = 4 \]
So, the intersection point is \( (2, 4) \).

### Step 7: Determine the feasible region
- The feasible region is bounded by the points (10, 0), (4, 0), (0, 5), (0, 8), and (2, 4).

### Conclusion
The solution set for the system of inequalities is the area bounded by these points, including the points on the lines since the inequalities are less than or equal to.

---