To solve the system of equations by elimination, we need to multiply one of the equations so that when added or subtracted, the $y$-terms cancel out.

The system of equations is:

1. $10x + 17y = 74$
2. $3x + y = 12$

We want to eliminate the $y$-terms. To do this, we need to make the coefficients of $y$ in both equations the same.

Looking at equation (2), the coefficient of $y$ is 1. To match it to the coefficient of $y$ in equation (1), which is 17, we can multiply equation (2) by 17. This will give us:

$17(3x + y) = 17(12)$ This simplifies to:

$51x + 17y = 204$

Now, we can use the system:

1. $10x + 17y = 74$
2. $51x + 17y = 204$

Next, subtract equation (1) from equation (2):

$(51x + 17y) - (10x + 17y) = 204 - 74$

The $17y$ terms will cancel out:

$(51x - 10x) = 130$

This simplifies to:

$41x = 130$

Solving for $x$:

$x = \frac{130}{41} = 3.17$

Therefore, the integer to multiply equation (2) by is 17.

Let me know if you'd like further clarification or if you'd like to continue solving the system! Here's a list of follow-up questions:

1. How can we solve for $y$ after finding $x$?
2. What is the geometric interpretation of a system of linear equations?
3. How does the substitution method differ from the elimination method?
4. What happens if the system has no solution or infinitely many solutions?
5. What is the significance of the coefficient of $y$ in elimination?

Tip: Always check your work by substituting your solution back into both original equations to ensure consistency.