Find the area of triangle formed by lines
`x - 2y = 5 and 2x + 3y = 10` with y-axis.

Answer

To find the area of the triangle formed by the lines \(x - 2y = 5\) and \(2x + 3y = 10\) with the y-axis, we can follow these steps:

### Step 1: Find the intersection points of the lines with the y-axis.

1. **For the line \(x - 2y = 5\)**:
- Set \(x = 0\):
\[ 0 - 2y = 5 \implies -2y = 5 \implies y = -\frac{5}{2} \]
- The intersection point is \( (0, -\frac{5}{2}) \).

2. **For the line \(2x + 3y = 10\)**:
- Set \(x = 0\):
\[ 2(0) + 3y = 10 \implies 3y = 10 \implies y = \frac{10}{3} \]
- The intersection point is \( (0, \frac{10}{3}) \).

### Step 2: Find the intersection point of the two lines.

1. **Solve the equations simultaneously**:
- From \(x - 2y = 5\), we can express \(x\) in terms of \(y\):
\[ x = 2y + 5 \]

- Substitute \(x\) in the second equation \(2x + 3y = 10\):
\[ 2(2y + 5) + 3y = 10 \implies 4y + 10 + 3y = 10 \implies 7y + 10 = 10 \implies 7y = 0 \implies y = 0 \]
- Substitute \(y = 0\) back into \(x = 2y + 5\):
\[ x = 2(0) + 5 = 5 \]
- The intersection point is \( (5, 0) \).

### Step 3: Identify the vertices of the triangle.

The vertices of the triangle formed by the lines and the y-axis are:
- \(A(0, -\frac{5}{2})\)
- \(B(0, \frac{10}{3})\)
- \(C(5, 0)\)

### Step 4: 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 of the triangle is the distance between points \(A\) and \(B\) on the y-axis:
\[ \text{Base} = \left| -\frac{5}{2} - \frac{10}{3} \right| = \left| -\frac{15}{6} - \frac{20}{6} \right| = \left| -\frac{35}{6} \right| = \frac{35}{6} \]

- The height of the triangle is the distance from point \(C(5, 0)\) to the y-axis, which is simply \(5\).

### Step 5: Substitute the values into the area formula.

\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times \frac{35}{6} \times 5 \]
\[ = \frac{1}{2} \times \frac{175}{6} = \frac{175}{12} \approx 14.58 \]

### Conclusion

The area of the triangle formed by the lines \(x - 2y = 5\) and \(2x + 3y = 10\) with the y-axis is approximately \(14.58\) square units.

---