Solve for X and Y: Exploring the System -8x + 5y = 2, -16x + 10y = 5

To solve this system of equations, we'll follow these steps:

• Step 1: Analyze the proportions of the coefficients to check for potential parallelism or redundancy.
• Step 2: Attempt elimination to directly identify any inconsistencies.

Now, let's perform the analysis:
Step 1: Notice the proportionality in both equations:
The first equation is $-8x + 5y = 2$ and the second is $-16x + 10y = 5$. Multiply the first equation by 2:
$-16x + 10y = 4$.
This equation is now equivalent in terms of $x$ and $y$ to the second equation, but with a different constant term.

Step 2: Subtract the modified first equation from the second equation:

This result, $0 = 1$, is a contradiction, indicating that the system is inconsistent.

Therefore, the system of equations has no solution.