Sort algebraic terms into groups based on like terms.

Problem Description:

The task is to sort the given algebraic terms into groups. The terms are as follows:

- \( 7xy \)
- \( 12xy \)
- \( 2x \)
- \( 3y^2 \)
- \( -11y^2 \)
- \( 7x^2 \)
- \( -4y \)
- \( -9x^2 \)
- \( 2x^2 \)
- \( 6y^2 \)
- \( 13x \)
- \( -5x \)
- \( 10y \)
- \( 2y \)

Solution Approach:

To solve this problem, we need to group the terms based on their like terms. Like terms are terms that have the same variables raised to the same powers. For example:
- \( 7x^2 \) and \( 2x^2 \) are like terms because they both have \( x^2 \).
- \( 3y^2 \), \( -11y^2 \), and \( 6y^2 \) are like terms because they all have \( y^2 \).

We will group the terms based on their variable structures:
1. Terms with \( x^2 \)
2. Terms with \( y^2 \)
3. Terms with \( xy \)
4. Terms with \( x \)
5. Terms with \( y \)

Step-by-Step Grouping:

#### 1. Terms with \( x^2 \):
- \( 7x^2 \)
- \( -9x^2 \)
- \( 2x^2 \)

These terms all have \( x^2 \) as their variable part.

#### 2. Terms with \( y^2 \):
- \( 3y^2 \)
- \( -11y^2 \)
- \( 6y^2 \)

These terms all have \( y^2 \) as their variable part.

#### 3. Terms with \( xy \):
- \( 7xy \)
- \( 12xy \)

These terms both have \( xy \) as their variable part.

#### 4. Terms with \( x \):
- \( 2x \)
- \( 13x \)
- \( -5x \)

These terms all have \( x \) as their variable part.

#### 5. Terms with \( y \):
- \( -4y \)
- \( 10y \)
- \( 2y \)

These terms all have \( y \) as their variable part.

Final Grouping:

Based on the above analysis, the terms can be grouped as follows:

1. Terms with \( x^2 \):
- \( 7x^2 \)
- \( -9x^2 \)
- \( 2x^2 \)

2. Terms with \( y^2 \):
- \( 3y^2 \)
- \( -11y^2 \)
- \( 6y^2 \)

3. Terms with \( xy \):
- \( 7xy \)
- \( 12xy \)

4. Terms with \( x \):
- \( 2x \)
- \( 13x \)
- \( -5x \)

5. Terms with \( y \):
- \( -4y \)
- \( 10y \)
- \( 2y \)

Final Answer:

\[
\boxed{
\begin{aligned}
&\text{Group 1 (} x^2 \text{): } 7x^2, -9x^2, 2x^2 \\
&\text{Group 2 (} y^2 \text{): } 3y^2, -11y^2, 6y^2 \\
&\text{Group 3 (} xy \text{): } 7xy, 12xy \\
&\text{Group 4 (} x \text{): } 2x, 13x, -5x \\
&\text{Group 5 (} y \text{): } -4y, 10y, 2y
\end{aligned}
}
\]