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The system of equations is:

\[
\begin{cases}
-x - 6y = -3 \\
2ax + 2y = -10
\end{cases}
\]

We need to determine the value of \(a\) such that the system has no solution. When a system of linear equations has no solution, it means the two lines represented by the equations are parallel. For two lines to be parallel, their slopes must be equal, but their constant terms must differ (inconsistent system).

Let's rewrite both equations in slope-intercept form (\(y = mx + b\)) to find the slopes:

1. First equation: 
\[
-x - 6y = -3 \implies -6y = x - 3 \implies y = -\frac{1}{6}x + \frac{1}{2}
\]
The slope of the first equation is \(m_1 = -\frac{1}{6}\).

2. Second equation:
\[
2ax + 2y = -10 \implies 2y = -2ax - 10 \implies y = -ax - 5
\]
The slope of the second equation is \(m_2 = -a\).

### Condition for no solution:
The system has no solution if the slopes are equal, but the constant terms are different. So, we set the slopes equal:
\[
-\frac{1}{6} = -a \implies a = \frac{1}{6}
\]

Thus, the value of \(a\) is \( \boxed{\frac{1}{6}} \), which corresponds to option B.

Would you like further explanation on how to recognize inconsistent systems, or have any additional questions?

Here are five related questions for practice:
1. How do you determine if a system of equations has infinitely many solutions?
2. What is the condition for two lines to be perpendicular?
3. How can you solve a system of equations using substitution?
4. What is the determinant of a 2x2 matrix and how does it relate to the solution of a system of equations?
5. How do you graph a system of linear equations to visually determine the number of solutions?

**Tip**: Always check the slopes when determining if a system has no solution. Equal slopes but different intercepts indicate parallel, non-intersecting lines.

If the system of equations -x - 6y = -3 and 2ax + 2y = -10 has no solution, find the value of a?

Solve the system of equations -x - 6y = -3 and 2ax + 2y = -10 by determining the value of 'a' for which the system has no solution. Learn key concepts about parallel lines, slopes, and inconsistent systems. Suitable for Grades 10-12.

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