The problem involves solving the system of equations:

1. $-6X + 5Y = 1$
2. $6X + 4Y = -10$

We will solve this using the substitution or elimination method.

Step 1: Eliminate $X$

To eliminate $X$, we add the two equations because the coefficients of $X$ ($-6$ and $6$) are opposites.

$(-6X + 5Y) + (6X + 4Y) = 1 + (-10)$

Simplify: $5Y + 4Y = -9$

$9Y = -9$

Solve for $Y$: $Y = -1$

Step 2: Substitute $Y = -1$ into one equation

Substitute $Y = -1$ into the first equation ($-6X + 5Y = 1$):

$-6X + 5(-1) = 1$

Simplify: $-6X - 5 = 1$

$-6X = 6$

$X = -1$

Final Answer:

$X = -1, \, Y = -1$

Would you like a detailed explanation of another solving method? Let me know if you have questions!

Related Questions:

1. How would you solve this system using the substitution method?
2. How can you verify the solution of a system of equations?
3. What happens if a system of equations has no solution?
4. What is the difference between consistent and inconsistent systems?
5. Can you graphically represent this system of equations?

Tip:

When solving systems of equations, always check your solution by substituting $X$ and $Y$ into both original equations!