The graph of ax+by+c=0 is

Options

A. a straight line parallel to x axis
B. a straight line parallel to y axis
C. a genearl straight line
D. name of these

Answer

To determine the nature of the graph represented by the equation \( ax + by + c = 0 \), we can analyze the equation step by step.

### Step-by-Step Solution:

1. **Identify the Equation**: The equation given is \( ax + by + c = 0 \). This is a linear equation in two variables, \( x \) and \( y \).

2. **Rearranging the Equation**: We can rearrange the equation to express \( y \) in terms of \( x \):
\[ by = -ax - c \]
Dividing both sides by \( b \) (assuming \( b \neq 0 \)):
\[ y = -\frac{a}{b}x - \frac{c}{b} \]
This is now in the slope-intercept form \( y = mx + b \), where \( m = -\frac{a}{b} \) is the slope and \( b = -\frac{c}{b} \) is the y-intercept.

3. **Understanding the Graph**: The equation \( y = mx + b \) represents a straight line in the Cartesian plane. The slope \( m \) indicates the steepness and direction of the line, while the y-intercept indicates where the line crosses the y-axis.

4. **Special Cases**:
- If \( a = 0 \): The equation simplifies to \( by + c = 0 \) or \( y = -\frac{c}{b} \), which is a horizontal line (parallel to the x-axis).
- If \( b = 0 \): The equation simplifies to \( ax + c = 0 \) or \( x = -\frac{c}{a} \), which is a vertical line (parallel to the y-axis).

5. **Conclusion**: Since the equation \( ax + by + c = 0 \) can represent different types of lines depending on the values of \( a \), \( b \), and \( c \), we conclude that it represents a general straight line in two-dimensional space.

### Final Answer:
The graph of the equation \( ax + by + c = 0 \) is a straight line in two dimensions.

---