To solve the system of equations using the elimination method, we are aiming to eliminate one of the variables by adding or subtracting the equations. Here's the system of equations:

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

Step 1: Make the coefficients of one of the variables the same in both equations

Let's eliminate $x$. To do this, we will multiply the second equation by 2 so that the coefficient of $x$ in both equations will match.

The second equation becomes: $2(x - 2y) = 2(-4)$ Simplifying that: $2x - 4y = -8$

Now, we have the system: $2x + 5y = 19 \quad \text{(Equation 1)}$ $2x - 4y = -8 \quad \text{(New Equation 2)}$

Step 2: Subtract the second equation from the first to eliminate $x$

Now subtract Equation 2 from Equation 1: $(2x + 5y) - (2x - 4y) = 19 - (-8)$ Simplifying the left side: $2x - 2x + 5y + 4y = 19 + 8$ $9y = 27$

Step 3: Solve for $y$

Now divide both sides by 9: $y = \frac{27}{9} = 3$

Step 4: Substitute $y = 3$ back into one of the original equations

Substitute $y = 3$ into the second equation $x - 2y = -4$: $x - 2(3) = -4$ $x - 6 = -4$ $x = -4 + 6 = 2$

Final Solution:

$x = 2, \quad y = 3$

Would you like any further explanation or clarification?

Here are some related questions you might find useful:

1. How do you use substitution instead of elimination to solve this system?
2. Can the elimination method be used for systems with three variables?
3. What happens if the system of equations has no solution or infinitely many solutions?
4. How can you check your solution for this system?
5. What are some real-life examples where solving systems of equations is useful?

Tip: When using the elimination method, make sure the coefficients of one of the variables are made the same in both equations. This will simplify the process of elimination.