Write the relationship between the coefficients , if the following pair of equations are 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 can follow these steps:

### Step 1: Understand the concept of inconsistent equations
Inconsistent equations are those that do not have any solution. This occurs when the two lines represented by the equations are parallel.

### Step 2: Write down the given equations
The two equations provided are:
1. \( ax + by + c = 0 \)
2. \( a'x + b'y + c' = 0 \)

### Step 3: Identify the coefficients
From the equations, we can identify the coefficients as follows:
- For the first equation:
- Coefficient of \( x \) is \( a \)
- Coefficient of \( y \) is \( b \)
- Constant term is \( c \)

- For the second equation:
- Coefficient of \( x \) is \( a' \)
- Coefficient of \( y \) is \( b' \)
- Constant term is \( c' \)

### Step 4: Set up the condition for inconsistency
For the two equations to be inconsistent (i.e., the lines are parallel), the ratios of the coefficients must be equal, but the ratio of the constants must not be equal. This gives us the following relationships:
\[ \frac{a}{a'} = \frac{b}{b'} \neq \frac{c}{c'} \]

### Step 5: State the relationship
Thus, the relationship between the coefficients for the equations to be inconsistent is:
- The coefficients \( a, b, c \) and \( a', b', c' \) must satisfy:
\[ \frac{a}{a'} = \frac{b}{b'} \quad \text{and} \quad \frac{c}{c'} \neq \frac{a}{a'} \]

### Summary
The relationship between the coefficients of the given pair of equations for them to be inconsistent is that the ratios of the coefficients of \( x \) and \( y \) must be equal, while the ratio of the constants must differ.

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