Find the equation of lines whose `X` and `Y`-intercepts are as follows :
`(i) 2` and `3` `(ii) `-2` and `-5`
`(iii) 3` and `-5` `(iv) 4` and `-2`

Answer

To find the equations of lines given their x and y intercepts, we can use the intercept form of the equation of a line, which is given by:

\[ \frac{x}{a} + \frac{y}{b} = 1 \]

where \(a\) is the x-intercept and \(b\) is the y-intercept.

Let's solve the given problems step by step.

### (i) X-intercept = 2 and Y-intercept = 3

1. **Identify the intercepts**: Here, \(a = 2\) and \(b = 3\).
2. **Substitute into the equation**:
\[ \frac{x}{2} + \frac{y}{3} = 1 \]
3. **Clear the denominators**: Multiply through by 6 (the least common multiple of 2 and 3):
\[ 6 \left(\frac{x}{2}\right) + 6 \left(\frac{y}{3}\right) = 6 \cdot 1 \]
This simplifies to:
\[ 3x + 2y = 6 \]

### (ii) X-intercept = -2 and Y-intercept = -5

1. **Identify the intercepts**: Here, \(a = -2\) and \(b = -5\).
2. **Substitute into the equation**:
\[ \frac{x}{-2} + \frac{y}{-5} = 1 \]
3. **Clear the denominators**: Multiply through by -10:
\[ -10 \left(\frac{x}{-2}\right) + -10 \left(\frac{y}{-5}\right) = -10 \cdot 1 \]
This simplifies to:
\[ 5x + 2y = 10 \]

### (iii) X-intercept = 3 and Y-intercept = -5

1. **Identify the intercepts**: Here, \(a = 3\) and \(b = -5\).
2. **Substitute into the equation**:
\[ \frac{x}{3} + \frac{y}{-5} = 1 \]
3. **Clear the denominators**: Multiply through by 15:
\[ 15 \left(\frac{x}{3}\right) + 15 \left(\frac{y}{-5}\right) = 15 \cdot 1 \]
This simplifies to:
\[ 5x - 3y = 15 \]

### (iv) X-intercept = 4 and Y-intercept = -2

1. **Identify the intercepts**: Here, \(a = 4\) and \(b = -2\).
2. **Substitute into the equation**:
\[ \frac{x}{4} + \frac{y}{-2} = 1 \]
3. **Clear the denominators**: Multiply through by 4:
\[ 4 \left(\frac{x}{4}\right) + 4 \left(\frac{y}{-2}\right) = 4 \cdot 1 \]
This simplifies to:
\[ x - 2y = 4 \]

### Final Answers
1. For (i): \(3x + 2y = 6\)
2. For (ii): \(5x + 2y = 10\)
3. For (iii): \(5x - 3y = 15\)
4. For (iv): \(x - 2y = 4\)