We are given the system of equations:

1. $2x - y = -6$
2. $5x + 3y = -4$

Step 1: Solve one equation for one variable

Let's solve the first equation for $y$:

$2x - y = -6$ Add $y$ to both sides:

$2x = y - 6$ Now solve for $y$:

$y = 2x + 6$

Step 2: Substitute $y$ in the second equation

Now substitute $y = 2x + 6$ into the second equation $5x + 3y = -4$:

$5x + 3(2x + 6) = -4$ Distribute the $3$ on the left-hand side:

$5x + 6x + 18 = -4$

Step 3: Solve for $x$

Combine like terms:

$11x + 18 = -4$ Subtract 18 from both sides:

$11x = -4 - 18$ $11x = -22$ Now divide by 11:

$x = \frac{-22}{11} = -2$

Step 4: Substitute $x = -2$ back into the expression for $y$

Substitute $x = -2$ into $y = 2x + 6$:

$y = 2(-2) + 6 = -4 + 6 = 2$

Final Answer:

The solution to the system of equations is: $x = -2, \quad y = 2$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How do you graphically represent the solution of a system of equations?
2. What happens if the system of equations has no solution?
3. Can you solve the system using the elimination method and compare the results?
4. How would you check if your solution satisfies both equations?
5. How can systems of equations be applied in real-world problems?

Tip: Always check your solution by substituting it back into the original equations!