The problem asks to solve the system of equations using the addition (or elimination) method. Here are the two equations:

1. $4x + 5y = -19$
2. $3x - 5y = -23$

To solve this system using the elimination method, we can add the two equations together in such a way that the $y$-terms cancel each other out.

Step-by-step solution:

1. Add the equations: $(4x + 5y) + (3x - 5y) = -19 + (-23)$ Simplifying: $4x + 3x = -42$ $7x = -42$ Now, solve for $x$: $x = \frac{-42}{7} = -6$

2. Substitute $x = -6$ into one of the original equations: Use the first equation $4x + 5y = -19$: $4(-6) + 5y = -19$ Simplifying: $-24 + 5y = -19$ Solving for $y$: $5y = -19 + 24 = 5$ $y = \frac{5}{5} = 1$

Final Answer:

$x = -6, \quad y = 1$

Would you like a further breakdown or have any additional questions? Here are some related questions you may find useful:

1. What would happen if you used substitution instead of elimination?
2. How can the elimination method be applied to a system with more variables?
3. What if one of the equations is in a different form, such as $2x + 3y = 7$?
4. How do you check the solution once you find values for $x$ and $y$?
5. What is the geometric interpretation of solving a system of linear equations?

Tip: Always ensure that the coefficients of one variable are opposites (or can be made opposites) when using the elimination method for simpler solutions.