Simplifying expressions worksheets | Worsheets library - Free Printable

Problem Analysis:

The task is to navigate through a maze by combining like terms in algebraic expressions. The goal is to start from the top of the maze and move to the bottom, selecting only those squares that contain expressions equivalent to the starting expression and the ending expression.

#### Key Steps:
1. Identify the Starting Expression: The starting expression at the top of the maze is \( 3n^2 + n \).
2. Identify the Ending Expression: The ending expression at the bottom of the maze is also \( 3n^2 + n \).
3. Combine Like Terms: For each square in the maze, simplify the given expression by combining like terms.
4. Find Equivalent Expressions: Move only to squares where the simplified expression matches either the starting or ending expression.
5. Trace the Path: Follow the path that leads from the top to the bottom using only valid squares.

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Solution:

#### Step 1: Simplify Each Expression
We will simplify each expression in the maze by combining like terms. Let's go row by row.

##### Top Row:
- \( 3n^2 + n \) (Already simplified)

##### Second Row:
- \( 2n^2 + n + m^2 \): Cannot be simplified further.
- \( n^2 + m + 2m^2 \): Cannot be simplified further.
- \( n^2 - n + 3n \): Combine like terms: \( n^2 + 2n \).
- \( 2n^2 + n^2 - m + n^2 \): Combine like terms: \( 5n^2 - m \).

##### Third Row:
- \( 4n^2 + n - m^2 \): Cannot be simplified further.
- \( 5n^2 + n - 2m - n \): Combine like terms: \( 5n^2 - 2m \).
- \( 2t + 3m^2 - t + n - t \): Combine like terms: \( 3m^2 + n \).
- \( 3m^2 - 6n^2 + n \): Cannot be simplified further.

##### Fourth Row:
- \( 3n^2 + n - n^2 \): Combine like terms: \( 2n^2 + n \).
- \( 3n^2 + n - 2n \): Combine like terms: \( 3n^2 - n \).
- \( mn - 3m^2 + 6mn - m \): Combine like terms: \( 7mn - 3m^2 - m \).
- \( 3m^2 + mn + n - mn \): Combine like terms: \( 3m^2 + n \).
- \( n^2 - 2m^2 + n \): Cannot be simplified further.

##### Fifth Row:
- \( 3n^2 - n + 2n \): Combine like terms: \( 3n^2 + n \).
- \( 3n^2 + 2mn + n - 2mn \): Combine like terms: \( 3n^2 + n \).
- \( 3n^2 + n^2 - mn^2 + n \): Combine like terms: \( 4n^2 - mn^2 + n \).
- \( mn + 3n^2 - m^2 + n \): Cannot be simplified further.
- \( mn - 4m^2 + 7m^2 - m + n \): Combine like terms: \( mn + 3m^2 - m + n \).

##### Sixth Row:
- \( 5n^2 + 2n - 2n^2 - n \): Combine like terms: \( 3n^2 + n \).
- \( 5n^2 + n + 2n \): Combine like terms: \( 5n^2 + 3n \).
- \( mn - 4m^2 + 7m^2 - m + n \): Combine like terms: \( mn + 3m^2 - m + n \).

##### Seventh Row:
- \( m^2 + n + 2m^3 \): Cannot be simplified further.
- \( 2n^3 + n - 2m^2 \): Cannot be simplified further.
- \( 3n^2 + n \): Already simplified.

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#### Step 2: Identify Valid Paths
We need to find a path from the top (\( 3n^2 + n \)) to the bottom (\( 3n^2 + n \)) using only squares that simplify to \( 3n^2 + n \).

##### Valid Squares:
- Top row: \( 3n^2 + n \)
- Second row: None
- Third row: None
- Fourth row: None
- Fifth row: \( 3n^2 + n \) (second square)
- Sixth row: \( 3n^2 + n \) (first square)
- Seventh row: \( 3n^2 + n \) (third square)

##### Path:
1. Start at \( 3n^2 + n \) (top).
2. Move down to \( 3n^2 + n \) (fifth row, second square).
3. Move down to \( 3n^2 + n \) (sixth row, first square).
4. Move down to \( 3n^2 + n \) (seventh row, third square).

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Final Answer:

The valid path through the maze is:
\[
\boxed{3n^2 + n \rightarrow 3n^2 + n \rightarrow 3n^2 + n \rightarrow 3n^2 + n}
\]