Question

In order to solve this system of equations using elimination/combination method the first thing Emma should do is
6x-3y=18 and -3x-9y=-93
Multiply the first equation by 3
By -2
Multiply the second equation by 2
By -3

Answer 1

To solve the system of equations using the elimination/combination method, we want to eliminate one of the variables by making their coefficients the same (with opposite signs).

The given equations are:
1. \(6x - 3y = 18\)
2. \(-3x - 9y = -93\)

### Step 1: Simplifying equations

First, let’s simplify the second equation by dividing everything by -3, which gives us a simpler form:

\[
x + 3y = 31 \quad (3)
\]

Now we have:
1. \(6x - 3y = 18\)
2. \(x + 3y = 31\)
3. Or, you can keep the original if you prefer.

### Step 2: Make coefficients of one variable match

We can eliminate \(y\) by manipulating both equations.

From equation (1), multiply the entire equation by 1 (to keep it the same for comparison):

\[
6x - 3y = 18 \quad (1)
\]

From equation (2) (the simplified version), multiply the entire equation by 3 to match the coefficient of \(y\):

\[
3(x + 3y) = 3(31) \rightarrow 3x + 9y = 93 \quad (4)
\]

### Step 3: Setting up for elimination

Now you have:
1. \(6x - 3y = 18 \) (equation 1)
2. \(3x + 9y = 93\) (equation 4)

You can now match the y-coefficients:

- We can multiply (1) by 3 to get:

\[
18x - 9y = 54 \quad (5)
\]

Now we have two equations we can work with:

1. \(18x - 9y = 54\)
2. \(3x + 9y = 93\)

### Step 4: Elimination

Now, we can add equations (5) and (4):

\[
(18x - 9y) + (3x + 9y) = 54 + 93
\]

The \(y\) terms cancel out, leading to:

\[
21x = 147
\]

Solving for \(x\):

\[
x = 7
\]

You can then substitute \(x = 7\) back into either equation to solve for \(y\).

---
### Conclusion
To directly answer your initial question, the best option would be to multiply the second equation by \(3\) in order to eliminate \(y\) effectively when adding the two equations.

So the answer is:
**Multiply the second equation by 3.**