Theequationoflineuis

Question

Theequationoflineuis
y
 = 
–7
5
x
 + 	9
.	 	The
equationoflinevis
y
 = 
7
5
x
 + 	
6
7
.	 	Are
lineuandlinevparallelorperpendicular?

GPT-4o mini · Answer

To determine whether lines $ u $ and $ v $ are parallel or perpendicular, we need to analyze their slopes. The general equation of a line can be written in the slope-intercept form $ y = mx + b $, where $ m $ is the slope. 1. For line $ u $: $$ y = -\frac{7}{5}x + 9 $$ The slope ($ m_u $) of line $ u $ is $ -\frac{7}{5} $. 2. For line $ v $: $$ y = \frac{7}{5}x + \frac{67}{1} $$ The slope ($ m_v $) of line $ v $ is $ \frac{7}{5} $. Now we compare the slopes: - Two lines are **parallel** if their slopes are equal, i.e., $ m_u = m_v $. - Two lines are **perpendicular** if the product of their slopes equals $-1$, i.e., $ m_u \times m_v = -1 $. Calculating the product of the slopes: $$ m_u \times m_v = \left(-\frac{7}{5}\right) \times \left(\frac{7}{5}\right) = -\frac{49}{25} $$ Since $ -\frac{49}{25} \neq -1 $, the lines are not perpendicular. Also, since $ -\frac{7}{5} \neq \frac{7}{5} $, the lines are not parallel either. ### Conclusion: Lines $ u $ and $ v $ are neither parallel nor perpendicular.