The area bounded by the lines `x = 0, y = 0, x + y = 1, 2 x + 3y = 6 ` (in square units ) is

Options

A. 2
B. `2 (1)/(3)`
C. `2 (1)/(2)`
D. 3

Answer

To find the area bounded by the lines \(x = 0\), \(y = 0\), \(x + y = 1\), and \(2x + 3y = 6\), we will follow these steps:

### Step 1: Identify the lines and their intersections
1. The lines are:
- \(x = 0\) (the y-axis)
- \(y = 0\) (the x-axis)
- \(x + y = 1\)
- \(2x + 3y = 6\)

### Step 2: Find the intersection points of the lines
2. **Find the intersection of \(x + y = 1\) and \(2x + 3y = 6\)**:
- From \(x + y = 1\), we can express \(y\) in terms of \(x\):
\[ y = 1 - x \]
- Substitute \(y\) into \(2x + 3y = 6\):
\[ 2x + 3(1 - x) = 6 \]
\[ 2x + 3 - 3x = 6 \]
\[ -x + 3 = 6 \implies -x = 3 \implies x = -3 \]
- Substitute \(x = -3\) back into \(y = 1 - x\):
\[ y = 1 - (-3) = 4 \]
- So, the intersection point is \((-3, 4)\).

3. **Find the intersection of \(x + y = 1\) with the axes**:
- Setting \(x = 0\):
\[ y = 1 \implies (0, 1) \]
- Setting \(y = 0\):
\[ x = 1 \implies (1, 0) \]

4. **Find the intersection of \(2x + 3y = 6\) with the axes**:
- Setting \(x = 0\):
\[ 3y = 6 \implies y = 2 \implies (0, 2) \]
- Setting \(y = 0\):
\[ 2x = 6 \implies x = 3 \implies (3, 0) \]

### Step 3: Identify the vertices of the bounded area
5. The vertices of the bounded area formed by the lines are:
- \(O(0, 0)\)
- \(A(1, 0)\)
- \(B(0, 1)\)
- \(C(0, 2)\)
- \(D(3, 0)\)

### Step 4: Calculate the area of the bounded region
6. **Divide the area into two triangles**:
- Triangle \(OAB\) with vertices \(O(0,0)\), \(A(1,0)\), \(B(0,1)\).
- Triangle \(OCD\) with vertices \(O(0,0)\), \(C(0,2)\), \(D(3,0)\).

7. **Calculate the area of triangle \(OAB\)**:
- The base is \(1\) (from \(O\) to \(A\)) and the height is \(1\) (from \(O\) to \(B\)):
\[ \text{Area}_{OAB} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \text{ square units} \]

8. **Calculate the area of triangle \(OCD\)**:
- The base is \(3\) (from \(O\) to \(D\)) and the height is \(2\) (from \(O\) to \(C\)):
\[ \text{Area}_{OCD} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 2 = 3 \text{ square units} \]

### Step 5: Find the area of the bounded region
9. **Total area of the bounded region**:
- The area of the region bounded by the lines is:
\[ \text{Area}_{bounded} = \text{Area}_{OCD} - \text{Area}_{OAB} = 3 - \frac{1}{2} = 2.5 \text{ square units} \]

### Final Answer
The area bounded by the lines is \(2.5\) square units.

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