A pair of linear equations is said to be inconsistent if it has:

Options

A. at least one solution
B. no solution
C. infinitely many solutions
D. unique solution

Answer

To determine when a pair of linear equations is inconsistent, we need to understand the conditions under which two lines represented by these equations do not intersect.

### Step-by-Step Solution:

1. **Understanding Linear Equations**:
A pair of linear equations can be represented in the standard form:
\[ a_1x + b_1y + c_1 = 0 \quad \text{(Equation 1)} \]
\[ a_2x + b_2y + c_2 = 0 \quad \text{(Equation 2)} \]

2. **Condition for Inconsistency**:
For the two lines represented by these equations to be inconsistent, they must be parallel. This means they will never intersect.

3. **Identifying Parallel Lines**:
Two lines are parallel if their slopes are equal but their y-intercepts are different. The slope of a line given by the equation \(Ax + By + C = 0\) is given by \(-\frac{A}{B}\). Therefore, for the two lines to be parallel:
\[ \frac{a_1}{b_1} = \frac{a_2}{b_2} \quad \text{(Equal slopes)} \]
However, the lines must not be the same, which means:
\[ \frac{c_1}{b_1} \neq \frac{c_2}{b_2} \quad \text{(Different y-intercepts)} \]

4. **Conclusion**:
If the above conditions are satisfied, the pair of linear equations is inconsistent, meaning there is no solution to the system of equations.

### Final Answer:
A pair of linear equations is said to be inconsistent if it has no solution, which occurs when the lines are parallel.

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