The system of equations given is:

$5x + y = 12$ $2x - 3y = 15$

We are tasked with solving this using the method of elimination.

Step 1: Align the equations

Let's write both equations clearly:

1. $5x + y = 12$
2. $2x - 3y = 15$

Step 2: Eliminate one variable

We want to eliminate one variable by making the coefficients of either $x$ or $y$ the same in both equations. Let's eliminate $y$.

To do this, we multiply the first equation by 3 to match the coefficient of $y$ in the second equation:

$3(5x + y) = 3(12)$ $15x + 3y = 36$

Now, the system of equations becomes:

1. $15x + 3y = 36$
2. $2x - 3y = 15$

Step 3: Add the two equations

Add both equations to eliminate $y$:

$(15x + 3y) + (2x - 3y) = 36 + 15$ $17x = 51$

Step 4: Solve for $x$

Now, solve for $x$:

$x = \frac{51}{17} = 3$

Step 5: Substitute $x = 3$ back into one of the original equations

Substitute $x = 3$ into the first equation:

$5(3) + y = 12$ $15 + y = 12$ $y = 12 - 15 = -3$

Solution:

The solution is $x = 3$ and $y = -3$, which corresponds to option A.

Do you want further details or have any questions?

Here are five related questions for further practice:

1. How do you choose which variable to eliminate in elimination method?
2. Can this system be solved using substitution method? If so, how?
3. What are the advantages of using elimination over substitution?
4. How would the process change if you needed to eliminate $x$ first?
5. Can the same system be solved graphically? How would the solution look?

Tip: Always check your solution by substituting the values back into the original equations.