The problem presents a system of linear equations:

$6x + 10y = -4$ $ax + 5y = 1$

The task is to find the value of $a$ such that the system has no solution.

Steps to solve:

1. Condition for no solution: For a system of two linear equations to have no solution, the lines represented by the equations must be parallel. Two lines are parallel if their slopes are equal, but their constants on the right side (after simplifying) are different.

2. Rewrite the equations in slope-intercept form (i.e., solve for $y$) to identify the slopes.

   From the first equation: $6x + 10y = -4 \implies 10y = -6x - 4 \implies y = -\frac{3}{5}x - \frac{2}{5}$ The slope of the first equation is $-\frac{3}{5}$.

   From the second equation: $ax + 5y = 1 \implies 5y = -ax + 1 \implies y = -\frac{a}{5}x + \frac{1}{5}$ The slope of the second equation is $-\frac{a}{5}$.

3. Set the slopes equal to ensure the lines are parallel: $-\frac{3}{5} = -\frac{a}{5}$ This simplifies to: $a = 3$

4. Check the constant terms: For the system to have no solution, the constant terms on the right-hand side of the equations must be different.

   • For the first equation: the constant term is $-\frac{2}{5}$.
   • For the second equation: the constant term is $\frac{1}{5}$.

Since these constants are different, the system has no solution when $a = 3$.

Final Answer:

The value of $a$ is $\boxed{3}$.

Would you like a detailed explanation of any steps?

Here are five related questions to expand on this:

1. How do you find the solution to a system of equations when there is a solution?
2. What happens when two lines have the same slope and the same intercept?
3. How do you interpret the geometrical meaning of a system with infinitely many solutions?
4. Can you explain how elimination or substitution methods are used to solve systems of equations?
5. What is the condition for two linear equations to represent the same line?

Tip: When solving systems of equations, always check both the slopes and the intercepts to determine whether the system has no solution, infinitely many solutions, or a unique solution.