To solve the system of equations given by:

1. $4x + 9y = -19$
2. $-4x - 7y = 13$

We can use the elimination or substitution method. Here, I'll use the elimination method.

Step 1: Add the equations

Adding both equations will eliminate $x$:

$(4x - 4x) + (9y - 7y) = -19 + 13$

This simplifies to:

$2y = -6$

Step 2: Solve for $y$

Dividing both sides by 2:

$y = -3$

Step 3: Substitute $y$ back into one of the original equations

Let's substitute $y = -3$ into the first equation:

$4x + 9(-3) = -19$

This simplifies to:

$4x - 27 = -19$

Step 4: Solve for $x$

Adding 27 to both sides:

$4x = 8$

Dividing by 4:

$x = 2$

Solution

The solution to the system of equations is:

$(x, y) = (2, -3)$

If you have any questions or need further details, feel free to ask! Here are some related questions you might find interesting:

1. How can you verify the solution using substitution?
2. What would happen if the equations represented parallel lines?
3. How can you graph these equations to visualize the solution?
4. What are the implications of having no solution in a system of equations?
5. How can systems of equations be solved using matrices?

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