## Exploring the Relationship between Lines and their Slopes: (k+1)x + 2y = 6 and m = k - 2

This article delves into the connection between the line represented by the equation **(k+1)x + 2y = 6** and its slope **m = k - 2**. We will explore how the value of **k** impacts the slope and, consequently, the orientation of the line.

### Understanding the Equation and Slope

The equation **(k+1)x + 2y = 6** represents a straight line in the coordinate plane. To understand the impact of **k** on the line, let's rewrite the equation in slope-intercept form (y = mx + c), where **m** is the slope and **c** is the y-intercept.

Solving for **y**:

**2y = - (k+1)x + 6****y = - (k+1)/2 x + 3**

Now we can clearly see that the slope of the line is **- (k+1)/2**.

### Analyzing the Relationship with m = k - 2

We are given that **m = k - 2**. Substituting this into the slope derived from the equation:

**- (k+1)/2 = k - 2**

Solving for **k**:

**-k - 1 = 2k - 4****3 = 3k****k = 1**

This means that for the slope **m** to be related to **k** through the equation **m = k - 2**, the value of **k** must be **1**.

### Implications for the Line

When **k = 1**, the equation becomes:

**2x + 2y = 6****y = -x + 3**

This means the line has a slope of **-1** and a y-intercept of **3**.

Therefore, only when **k = 1** does the line defined by the equation **(k+1)x + 2y = 6** have a slope that satisfies the relationship **m = k - 2**.

### Conclusion

By analyzing the equation and the relationship between **k** and **m**, we have determined that the line represented by **(k+1)x + 2y = 6** only satisfies the condition **m = k - 2** when **k = 1**. This understanding allows us to predict the slope of the line based on the value of **k** and understand how changing the value of **k** affects the line's orientation.