Which one of the following values of x,y satisfies the equations `2x + 3y le 6 : x ge 0, y ge 0`?

Options

A. x = 0, y = 3
B. x = 1, y = 2
C. x = 1, y = 1
D. x = 4, y = 0

Answer

To solve the problem of determining which values of \( x \) and \( y \) satisfy the inequalities \( 2x + 3y \leq 6 \), \( x \geq 0 \), and \( y \geq 0 \), we can follow these steps:

### Step 1: Identify the Boundary Line
The first step is to rewrite the inequality \( 2x + 3y \leq 6 \) as an equation to find the boundary line:
\[ 2x + 3y = 6 \]

### Step 2: Find Intercepts
Next, we find the x-intercept and y-intercept of the line:
- **For the x-intercept** (set \( y = 0 \)):
\[ 2x + 3(0) = 6 \implies 2x = 6 \implies x = 3 \]
So, the x-intercept is \( (3, 0) \).

- **For the y-intercept** (set \( x = 0 \)):
\[ 2(0) + 3y = 6 \implies 3y = 6 \implies y = 2 \]
So, the y-intercept is \( (0, 2) \).

### Step 3: Graph the Inequality
Now, we graph the line \( 2x + 3y = 6 \) and shade the region that satisfies \( 2x + 3y \leq 6 \). The shaded region will be below the line, including the line itself.

### Step 4: Check the Given Points
Now we will check each of the given options to see if they lie within the shaded region:

1. **Option A: \( (0, 3) \)**
\[ 2(0) + 3(3) = 9 \quad (9 \not\leq 6) \quad \text{(Not satisfied)} \]

2. **Option B: \( (1, 2) \)**
\[ 2(1) + 3(2) = 2 + 6 = 8 \quad (8 \not\leq 6) \quad \text{(Not satisfied)} \]

3. **Option C: \( (1, 1) \)**
\[ 2(1) + 3(1) = 2 + 3 = 5 \quad (5 \leq 6) \quad \text{(Satisfied)} \]

4. **Option D: \( (4, 0) \)**
\[ 2(4) + 3(0) = 8 \quad (8 \not\leq 6) \quad \text{(Not satisfied)} \]

### Conclusion
The only point that satisfies the inequalities is \( (1, 1) \). Therefore, the correct answer is:

**Option C: \( x = 1, y = 1 \)**.

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