The system of equations given in the image is:

$-6x + 4y = 26$ $5x + 2y = 37$

To eliminate one of the variables using the elimination method, we need to manipulate the equations so that either the $x$-terms or the $y$-terms become opposites.

Step 1: Focus on the coefficients of either $x$ or $y$.

Let's focus on eliminating $y$, since the coefficients of $y$ are $4$ and $2$. To eliminate $y$, we want to make the coefficients of $y$ in both equations equal but with opposite signs.

Step 2: Make the $y$-terms opposites.

To do this, we can multiply the second equation by $-2$, so that the coefficient of $y$ in the second equation becomes $-4$, which is the opposite of the coefficient of $y$ in the first equation.

Multiplying the second equation by $-2$:

$-2(5x + 2y = 37) \implies -10x - 4y = -74$

Now, the system becomes:

$-6x + 4y = 26$ $-10x - 4y = -74$

Step 3: Add the equations.

Now, add the two equations together:

$(-6x + 4y) + (-10x - 4y) = 26 + (-74)$

This simplifies to:

$-16x = -48$

Solving for $x$:

$x = \frac{-48}{-16} = 3$

Correct Answer:

The operation needed to eliminate one of the variables is Option C: Multiply the second equation by -2.

Would you like further details on solving the full system, or have any other questions?

Here are 5 related questions to explore:

1. How can we solve for $y$ after finding $x = 3$?
2. What would happen if we tried eliminating $x$ instead of $y$?
3. Why is multiplying one of the equations necessary for elimination?
4. Can you always use the elimination method, or are there cases where it doesn't work well?
5. How does elimination compare to the substitution method?

Tip: When using the elimination method, always focus on which variable can be easily eliminated by adjusting the coefficients.