A triangle is formed by the straight lines `x + 2y - 3 =0 , 3x - 2y + 7 =0 and y+ 1 =0 `, find graphically
the area of the triangle.

Answer

To find the area of the triangle formed by the lines \(x + 2y - 3 = 0\), \(3x - 2y + 7 = 0\), and \(y + 1 = 0\) graphically, we will follow these steps:

### Step 1: Find the Intersection Points of the Lines
We need to determine the points where the lines intersect to form the vertices of the triangle.

1. **Intersection of Line 1 and Line 2:**
- Line 1: \(x + 2y - 3 = 0\)
- Line 2: \(3x - 2y + 7 = 0\)

Substitute \(x\) from Line 1 into Line 2:
\[ x = 3 - 2y \quad \text{(from Line 1)} \]
Substitute into Line 2:
\[ 3(3 - 2y) - 2y + 7 = 0 \]
\[ 9 - 6y - 2y + 7 = 0 \]
\[ 16 - 8y = 0 \implies 8y = 16 \implies y = 2 \]
Substitute \(y = 2\) back into Line 1:
\[ x + 2(2) - 3 = 0 \implies x + 4 - 3 = 0 \implies x = -1 \]
So, the intersection point \(A\) is \((-1, 2)\).

2. **Intersection of Line 2 and Line 3:**
- Line 3: \(y + 1 = 0 \implies y = -1\)

Substitute \(y = -1\) into Line 2:
\[ 3x - 2(-1) + 7 = 0 \implies 3x + 2 + 7 = 0 \implies 3x + 9 = 0 \implies x = -3 \]
So, the intersection point \(B\) is \((-3, -1)\).

3. **Intersection of Line 1 and Line 3:**
Substitute \(y = -1\) into Line 1:
\[ x + 2(-1) - 3 = 0 \implies x - 2 - 3 = 0 \implies x = 5 \]
So, the intersection point \(C\) is \((5, -1)\).

### Step 2: Plot the Points
The vertices of the triangle are:
- \(A(-1, 2)\)
- \(B(-3, -1)\)
- \(C(5, -1)\)

### Step 3: Calculate the Area of the Triangle
The area \(A\) of a triangle formed by vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates of points \(A\), \(B\), and \(C\):
\[ \text{Area} = \frac{1}{2} \left| -1(-1 + 1) + (-3)(-1 - 2) + 5(2 + 1) \right| \]
\[ = \frac{1}{2} \left| 0 + 9 + 15 \right| = \frac{1}{2} \left| 24 \right| = 12 \text{ square units} \]

### Final Answer
The area of the triangle formed by the given lines is **12 square units**.