Find the equation of the line passing through the following points :
`(i) (1,2)` and `(4,7)`
`(ii) (-3,1)` and `(0,3)`
`(iii)` origin and `(1,4)`
`(iv) (-2,-3)` and `(1,2)`

Answer

To find the equation of a line passing through two points, we can use the point-slope form of the equation of a line, which is given by:

\[ y - y_1 = m(x - x_1) \]

where \( m \) is the slope of the line calculated as:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's solve the given problems step by step.

### (i) Points: (1, 2) and (4, 7)

**Step 1**: Identify the points and assign coordinates.
- Let \( (x_1, y_1) = (1, 2) \)
- Let \( (x_2, y_2) = (4, 7) \)

**Step 2**: Calculate the slope \( m \).
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 2}{4 - 1} = \frac{5}{3} \]

**Step 3**: Use the point-slope form.
\[ y - 2 = \frac{5}{3}(x - 1) \]

**Step 4**: Rearranging the equation:
\[ y - 2 = \frac{5}{3}x - \frac{5}{3} \]
\[ y = \frac{5}{3}x - \frac{5}{3} + 2 \]
\[ y = \frac{5}{3}x + \frac{1}{3} \]

**Step 5**: Convert to standard form:
\[ 5x - 3y + 1 = 0 \]

### (ii) Points: (-3, 1) and (0, 3)

**Step 1**: Identify the points.
- Let \( (x_1, y_1) = (-3, 1) \)
- Let \( (x_2, y_2) = (0, 3) \)

**Step 2**: Calculate the slope \( m \).
\[ m = \frac{3 - 1}{0 + 3} = \frac{2}{3} \]

**Step 3**: Use the point-slope form.
\[ y - 1 = \frac{2}{3}(x + 3) \]

**Step 4**: Rearranging the equation:
\[ y - 1 = \frac{2}{3}x + 2 \]
\[ y = \frac{2}{3}x + 3 \]

**Step 5**: Convert to standard form:
\[ 2x - 3y + 9 = 0 \]

### (iii) Points: Origin (0, 0) and (1, 4)

**Step 1**: Identify the points.
- Let \( (x_1, y_1) = (0, 0) \)
- Let \( (x_2, y_2) = (1, 4) \)

**Step 2**: Calculate the slope \( m \).
\[ m = \frac{4 - 0}{1 - 0} = 4 \]

**Step 3**: Use the point-slope form.
\[ y - 0 = 4(x - 0) \]

**Step 4**: Rearranging the equation gives:
\[ y = 4x \]

**Step 5**: Convert to standard form:
\[ 4x - y = 0 \]

### (iv) Points: (-2, -3) and (1, 2)

**Step 1**: Identify the points.
- Let \( (x_1, y_1) = (-2, -3) \)
- Let \( (x_2, y_2) = (1, 2) \)

**Step 2**: Calculate the slope \( m \).
\[ m = \frac{2 + 3}{1 + 2} = \frac{5}{3} \]

**Step 3**: Use the point-slope form.
\[ y + 3 = \frac{5}{3}(x + 2) \]

**Step 4**: Rearranging the equation:
\[ y + 3 = \frac{5}{3}x + \frac{10}{3} \]
\[ y = \frac{5}{3}x + \frac{10}{3} - 3 \]
\[ y = \frac{5}{3}x - \frac{5}{3} \]

**Step 5**: Convert to standard form:
\[ 5x - 3y - 5 = 0 \]

### Summary of Equations:
1. \( 5x - 3y + 1 = 0 \)
2. \( 2x - 3y + 9 = 0 \)
3. \( 4x - y = 0 \)
4. \( 5x - 3y - 5 = 0 \)