Which statement about the following equations is correct?

The equations are independent because the lines intersect in one point.
The equations are independent because the equations represent parallel lines.
The equations are dependent because the lines do not intersect.
The equations are dependent because the lines are the same line.

Question

Which statement about the following equations is correct?

The equations are independent because the lines intersect in one point.
The equations are independent because the equations represent parallel lines.
The equations are dependent because the lines do not intersect.
The equations are dependent because the lines are the same line.

UpStudy AI · Accepted Answer

The correct statement is:

**The equations are independent because the lines intersect in one point.**

**Explanation:**

Let's analyze the two given equations:

1. $$ y = x - 6 $$  
   - **Slope (m₁):** 1  
   - **y-intercept (b₁):** -6

2. $$ y = -x - 2 $$  
   - **Slope (m₂):** -1  
   - **y-intercept (b₂):** -2

**Key Points:**

- **Slopes:** The slopes of the two lines are different ($$ m₁ = 1 $$ and $$ m₂ = -1 $$). This means the lines are **not parallel** and will intersect at exactly one point.
  
- **Intersection:** Since the lines intersect at one distinct point, the system of equations has a **unique solution**. In the context of systems of equations:
  - **Independent:** The system has exactly one solution (the lines intersect at one point).
  - **Dependent:** The system has infinitely many solutions (the lines are identical).

**Why Other Options Are Incorrect:**

- **Parallel Lines:** If the lines were parallel (same slope but different y-intercepts), they would not intersect, and the system would have no solution. However, this is not the case here because the slopes are different.
  
- **Same Line:** If both equations represented the same line (identical slopes and y-intercepts), the system would have infinitely many solutions, making it dependent. Again, this is not applicable here.

Therefore, since the lines intersect at exactly one point, the system is **independent**.

**Answer:**  
The equations are independent because the lines intersect in one point.
The equations are independent because the lines intersect at one point.

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