If a pair of linear equations in two variables is consistent, then the lines represented by two equations are

Options

A. parallel
B. always coincident
C. intersecting or coincident
D. always intersecting

Explanation

To solve the question, we need to analyze the properties of linear equations in two variables and what it means for them to be consistent.

### Step-by-Step Solution:

1. **Understanding Consistency**:
- A pair of linear equations in two variables is said to be consistent if there is at least one solution to the equations. This means that the lines represented by these equations intersect at least at one point.

**Hint**: Remember that consistent equations have at least one solution.

2. **Types of Lines**:
- There are three possible scenarios for two lines represented by linear equations:
- **Parallel Lines**: These lines never intersect and thus have no solutions. Therefore, they are inconsistent.
- **Coincident Lines**: These lines lie on top of each other, meaning they have infinitely many solutions (every point on the line is a solution).
- **Intersecting Lines**: These lines cross each other at exactly one point, which means they have one unique solution.

**Hint**: Visualize the three types of lines: parallel, coincident, and intersecting.

3. **Analyzing Each Case**:
- **Parallel Lines**: Since they do not intersect, they do not provide any solution. Hence, they cannot be consistent.
- **Coincident Lines**: Since they overlap completely, they have infinitely many solutions. Thus, they are consistent.
- **Intersecting Lines**: Since they intersect at one point, they have one solution. Thus, they are also consistent.

**Hint**: Check if the lines intersect or overlap to determine if they are consistent.

4. **Conclusion**:
- Therefore, if a pair of linear equations in two variables is consistent, the lines represented by these equations must be either intersecting or coincident. They cannot be parallel.

**Final Statement**: Hence, the lines represented by the two equations are either intersecting or coincident.

### Summary:
If a pair of linear equations in two variables is consistent, then the lines represented by the two equations are either intersecting or coincident.