Sort the systems of equations by their appropriate elimination method (addition or subtraction) for finding their solutions.

Steps to Solve

1. Understand the Elimination Method

The elimination method involves adding or subtracting two equations in a system to eliminate one of the variables. The choice between addition and subtraction depends on the coefficients of the variables in the equations.

2. Analyze when to Use Addition

Addition is used when the coefficients of one of the variables are opposites (e.g., $g$ and $-g$). Adding the equations will eliminate this variable.

3. Analyze when to Use Subtraction

Subtraction is used when the coefficients of one of the variables are the same (e.g., $4x$ and $4x$, or $2b$ and $2b$). Subtracting the equations will eliminate this variable. Alternatively, if we have a case like $4x$ and $-4x$, we can just add the two equations.

4. Sort the Systems of Equations

Now, let's sort the systems in the "Answer Bank":

• System 1: $2f + g = 3$ and $3f - g = 4$. Here, the coefficients of $g$ are $1$ and $-1$, which are opposites. Thus, we should use addition to eliminate $g$.
• System 2: $3a + 2b = 5$ and $a + 2b = 6$. Here, the coefficients of $b$ are both $2$. Thus, we should use subtraction to eliminate $b$.
• System 3: $4x + 3y = 3$ and $-4x + 2y = 2$. Here, the coefficients of $x$ are $4$ and $-4$, which are opposites. Thus, we should use addition to eliminate $x$.
• System 4: $6m - 5n = 8$ and $-2m - 5n = 6$. Here, the coefficients of $n$ are the same, $-5$. Thus, we should use subtraction to eliminate $n$.

5. Categorize Systems

Based on the above analysis, we can categorize the systems as follows:

• Elimination Using Addition:
  • $2f + g = 3$ $3f - g = 4$
  • $4x + 3y = 3$ $-4x + 2y = 2$
• Elimination Using Subtraction:
  • $3a + 2b = 5$ $a + 2b = 6$
  • $6m - 5n = 8$ $-2m - 5n = 6$