Let's break down the problem you've provided.

You are given the system of equations:

1. $x + y = 4$
2. $\frac{1}{2}x - 7y = 17$

The task is to figure out what constant needs to be multiplied to eliminate one of the variables when the equations are added.

Step 1: Eliminate $y$

To eliminate $y$, the coefficients of $y$ in both equations must be opposites of each other. Let's consider the coefficient of $y$ in both equations:

• The first equation has $y$ with a coefficient of 1 (as $x + y = 4$).
• The second equation has $y$ with a coefficient of -7 (as $\frac{1}{2}x - 7y = 17$).

To make the coefficients of $y$ opposites, we can multiply the first equation by 7, which will give us:

$7(x + y) = 7(4)$ $7x + 7y = 28$

Now, we have the system:

1. $7x + 7y = 28$
2. $\frac{1}{2}x - 7y = 17$

Step 2: Add the equations

Now, adding the two equations:

$7x + 7y + \frac{1}{2}x - 7y = 28 + 17$

Notice that $7y$ and $-7y$ cancel out, leaving us with:

$7x + \frac{1}{2}x = 45$

This is the simplified equation with $x$ as the only variable.

Conclusion:

To eliminate $y$, we needed to multiply the first equation by 7. The result of this multiplication was:

$7x + 7y = 28$

This matches answer choice a. the first equation to eliminate y.

Let me know if you need further clarifications or would like to go over the next questions!

Here are five questions related to this topic:

1. What method can you use to eliminate $x$ instead of $y$ in this system?
2. How do you solve for $x$ once the equation is simplified to $7x + \frac{1}{2}x = 45$?
3. What is the process for solving a system of equations using substitution?
4. Can this system be solved using graphing instead of elimination? How?
5. How does multiplying the second equation help with elimination?

Tip: When using the elimination method, always check that the coefficients of one of the variables match (in magnitude but are opposite signs) before adding the equations together.