1. **State the problem:** We are given the equation of line $v$ as $ax - 6y + 1 = 0$, where $a \in \mathbb{R}$. We know line $t$ is parallel to line $v$. We need to find the value of $a$. 2. **Recall the rule for parallel lines:** Two lines are parallel if and only if their slopes are equal. 3. **Find the slope of line $v$:** Rewrite the equation $ax - 6y + 1 = 0$ in slope-intercept form $y = mx + b$. $$ax - 6y + 1 = 0 \implies -6y = -ax - 1 \implies y = \frac{a}{6}x + \frac{1}{6}$$ So, the slope of line $v$ is $m_v = \frac{a}{6}$. 4. **Find the slope of line $t$:** Since line $t$ is parallel to line $v$, it must have the same slope. If line $t$ is given or implied to have a known slope, set that equal to $\frac{a}{6}$ to solve for $a$. 5. **Conclusion:** Without additional information about line $t$, such as its slope or equation, we cannot find a unique value for $a$. However, if line $t$ is parallel to $v$, then the slope of $t$ equals $\frac{a}{6}$. If the slope of line $t$ is known, say $m_t$, then: $$m_t = \frac{a}{6} \implies a = 6m_t$$ **Final answer:** The value of $a$ is $6$ times the slope of line $t$. If the problem provides the slope of line $t$, substitute it here to find $a$.