The equation of line \( v \) is: \( a x-6 y+1=0 \), where \( a \in \mathbb{R} \). Line \( t \) is parallel to line \( v \). Find the value of \( a \).

To find the value of \( a \), we start by rewriting the given equation of line \( v \) in slope-intercept form (i.e., \( y = mx + b \)), where \( m \) is the slope. Starting with the equation \( a x - 6 y + 1 = 0 \), we can isolate \( y \): \[ -6y = -ax - 1 \implies y = \frac{a}{6}x + \frac{1}{6} \] From this, we can see that the slope \( m \) of line \( v \) is \( \frac{a}{6} \). Since line \( t \) is parallel to line \( v \), it must have the same slope. Thus, line \( t \) will have the slope \( \frac{a}{6} \). For the two lines to be parallel: 1. We need to ensure the coefficients of \( x \) and \( y \) in the original form of \( v \) create a feasible slope that allows \( t \) to exist. A parallel line generally means they have the same orientation, thus the desired value for \( a \) can be any real number. Thus, \( a \) can take any real value \( a \in \mathbb{R} \). So, there’s no specific value for \( a \)—it can be any real number!