Question:

Let the equation of line L be \(ax + by = c\). The slope of line L is given by \(m_L = -\frac{a}{b}\), provided \(b \neq 0\). If \(b=0\), L is a vertical line with an undefined slope. **1. Determine the equation and slope of line M.** Line M is the reflection of L through the y-axis. To find the equation of M, replace \(x\) with \(-x\) in the equation of L: $$a(-x) + by = c$$ $$-ax + by = c$$ To find the slope of M, rearrange the equation into slope-intercept form \(y = mx + k\): $$by = ax + c$$ $$y = \frac{a}{b}x + \frac{c}{b}$$ The slope of M is \(m_M = \frac{a}{b}\), provided \(b \neq 0\). **2. Determine the equation and slope of line N.** Line N is the reflection of L through the x-axis. To find the equation of N, replace \(y\) with \(-y\) in the equation of L: $$ax + b(-y) = c$$ $$ax - by = c$$ To find the slope of N, rearrange the equation into slope-intercept form \(y = mx + k\): $$-by = -ax + c$$ $$y = \frac{a}{b}x - \frac{c}{b}$$ The slope of N is \(m_N = \frac{a}{b}\), provided \(b \neq 0\). **3. Compare the slopes of M and N.** From the derivations, the slope of M is \(m_M = \frac{a}{b}\) and the slope of N is \(m_N = \frac{a}{b}\). Thus, \(m_M = m_N\). This holds true when \(b \neq 0\). If \(b=0\), line L is \(ax=c\), which is a vertical line. In this case: Line M: \(-ax=c\), which is also a vertical line. Line N: \(ax=c\), which is also a vertical line. Vertical lines have undefined slopes, but they are parallel, meaning their slopes are considered equal in a geometric context. Therefore, the slopes of M and N are equal for all choices of a, b, and c.