Find the equation of a line passing through origin and parallel to the line joining the points `(1,3)` and `(2,-1)`.

Answer

To find the equation of a line passing through the origin and parallel to the line joining the points (1, 3) and (2, -1), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**: We are given two points, A(1, 3) and B(2, -1).

2. **Calculate the Slope of the Line AB**: The slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]
Substituting the coordinates of points A and B:
\[ m = \frac{-1 - 3}{2 - 1} = \frac{-4}{1} = -4 \]

3. **Use the Slope for the Parallel Line**: Since the line we want to find is parallel to line AB, it will have the same slope. Therefore, the slope of our desired line is also -4.

4. **Equation of the Line through the Origin**: The general equation of a line in slope-intercept form is:
\[ y = mx + c \]
Since the line passes through the origin (0, 0), we know that c = 0. Thus, the equation simplifies to:
\[ y = -4x \]

5. **Rearranging the Equation**: We can rearrange this equation into standard form:
\[ 4x + y = 0 \]

### Final Answer:
The equation of the line passing through the origin and parallel to the line joining the points (1, 3) and (2, -1) is:
\[ 4x + y = 0 \]