Write the relationship between the coefficients, if the following pair of equations is inconsistent.

ax + by + c = 0,
a'x + b'y + c' = 0

Answer

To determine the relationship between the coefficients of the given pair of equations when they are inconsistent, we need to analyze the conditions for inconsistency in linear equations.

### Step-by-Step Solution:

1. **Understand the Equations**:
We have two linear equations:
\[ ax + by + c = 0 \quad \text{(1)} \]
\[ a'x + b'y + c' = 0 \quad \text{(2)} \]

2. **Identify the Condition for Inconsistency**:
The equations are inconsistent if they represent parallel lines. For two lines to be parallel, their slopes must be equal, but their y-intercepts must be different.

3. **Find the Slopes**:
The slope of the first equation (1) can be found by rearranging it into the slope-intercept form \(y = mx + b\):
\[ by = -ax - c \implies y = -\frac{a}{b}x - \frac{c}{b} \]
Thus, the slope \(m_1 = -\frac{a}{b}\).

Similarly, for the second equation (2):
\[ b'y = -a'x - c' \implies y = -\frac{a'}{b'}x - \frac{c'}{b'} \]
Thus, the slope \(m_2 = -\frac{a'}{b'}\).

4. **Set the Slopes Equal**:
For the lines to be parallel:
\[ m_1 = m_2 \implies -\frac{a}{b} = -\frac{a'}{b'} \]
This simplifies to:
\[ \frac{a}{b} = \frac{a'}{b'} \quad \text{(3)} \]

5. **Condition for Different Intercepts**:
For the lines to be inconsistent, the y-intercepts must be different. This means:
\[ \frac{c}{b} \neq \frac{c'}{b'} \quad \text{(4)} \]

6. **Combine the Relationships**:
From (3) and (4), we can conclude that:
\[ \frac{a}{a'} = \frac{b}{b'} \quad \text{and} \quad \frac{c}{c'} \neq \frac{b}{b'} \]

### Final Relationship:
The relationship between the coefficients for the equations to be inconsistent is:
\[ \frac{a}{a'} = \frac{b}{b'} \quad \text{and} \quad \frac{c}{c'} \neq \frac{b}{b'} \]