x%2B2y%3D6-m%3Dk-2.jpeg "(k+1)x+2y=6 M=k-2")
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.