Question

What is the general equation of the line passes through the origin and (2,1)?

Answer 1

To find the general equation of a line passing through two points, we can use the formula for the slope-intercept form of a line, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

In this case, the line passes through the origin (0, 0) and the point (2, 1):

1. **Calculate the slope \( m \)**:
The slope \( m \) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (2, 1) \):
\[
m = \frac{1 - 0}{2 - 0} = \frac{1}{2}
\]

2. **Write the equation using the slope-intercept form**:
Since the line passes through the origin, the y-intercept \( b = 0 \). Thus, the equation of the line becomes:
\[
y = \frac{1}{2}x + 0
\]
This simplifies to:
\[
y = \frac{1}{2}x
\]

3. **General form of the equation**:
To write it in general form \( Ax + By + C = 0 \), we can rearrange the equation:
\[
\frac{1}{2}x - y = 0
\]
Multiplying through by 2 to eliminate the fraction gives:
\[
x - 2y = 0
\]
Therefore, the general equation of the line that passes through the origin and the point (2, 1) is:
\[
x - 2y = 0
\]