Solve graphically the following equations,
`x+2y=4,3x-2y=4`
Take 2 cm=1 unit on each axis.
Also find the area of the triangle formed by the lines and the x-axis.

Answer

To solve the equations \(x + 2y = 4\) and \(3x - 2y = 4\) graphically, we will follow these steps:

### Step 1: Convert the equations to slope-intercept form (y = mx + b)

1. **For the first equation:**
\[ x + 2y = 4 \implies 2y = 4 - x \implies y = \frac{4 - x}{2} = 2 - \frac{1}{2}x \]
This gives us the equation \(y = 2 - \frac{1}{2}x\).

2. **For the second equation:**
\[ 3x - 2y = 4 \implies -2y = 4 - 3x \implies 2y = 3x - 4 \implies y = \frac{3x - 4}{2} = \frac{3}{2}x - 2 \]
This gives us the equation \(y = \frac{3}{2}x - 2\).

### Step 2: Find points for each line

**For the first equation \(y = 2 - \frac{1}{2}x\):**
- When \(x = 0\):
\[ y = 2 - \frac{1}{2}(0) = 2 \quad \Rightarrow \quad (0, 2) \]
- When \(x = 2\):
\[ y = 2 - \frac{1}{2}(2) = 1 \quad \Rightarrow \quad (2, 1) \]
- When \(x = 4\):
\[ y = 2 - \frac{1}{2}(4) = 0 \quad \Rightarrow \quad (4, 0) \]

**For the second equation \(y = \frac{3}{2}x - 2\):**
- When \(x = 0\):
\[ y = \frac{3}{2}(0) - 2 = -2 \quad \Rightarrow \quad (0, -2) \]
- When \(x = 2\):
\[ y = \frac{3}{2}(2) - 2 = 1 \quad \Rightarrow \quad (2, 1) \]
- When \(x = 4\):
\[ y = \frac{3}{2}(4) - 2 = 4 \quad \Rightarrow \quad (4, 4) \]

### Step 3: Plot the points and draw the lines

- Plot the points from both equations on a graph with \(2 \text{ cm} = 1 \text{ unit}\).
- For the first equation, plot the points \((0, 2)\), \((2, 1)\), and \((4, 0)\).
- For the second equation, plot the points \((0, -2)\), \((2, 1)\), and \((4, 4)\).
- Draw straight lines through the points for each equation.

### Step 4: Find the intersection point

The lines intersect at the point \((2, 1)\).

### Step 5: Identify the triangle formed by the lines and the x-axis

The triangle is formed by the points:
- A: \((4, 0)\) (x-intercept of the first line)
- B: \((2, 1)\) (intersection point)
- C: \((0, -2)\) (y-intercept of the second line)

### Step 6: Calculate the area of the triangle

The area \(A\) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
- The base is the distance along the x-axis between points A and C. The x-coordinates of A and C are \(4\) and \(0\), respectively, so the base is \(4\) units.
- The height is the y-coordinate of point B, which is \(1\) unit.

Thus, the area is:
\[ A = \frac{1}{2} \times 4 \times 1 = 2 \text{ square units} \]

### Summary of the Solution:
- The intersection point of the lines is \((2, 1)\).
- The area of the triangle formed by the lines and the x-axis is \(2\) square units.