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The problem involves decrypting a hidden word based on specific coded values assigned to letters. Here’s a detailed explanation of how to approach it:

### Problem Summary:
Each letter of the alphabet is coded with an alternating pattern of positive and negative integers:
- \(1 \to A\)
- \(-2 \to B\)
- \(3 \to C\)
- \(-4 \to D\), and so on.

There are colored circles representing different numbers associated with letters. The problem provides the following equations:
1. \(\textcolor{red}{\text{Red}} \times \textcolor{green}{\text{Green}} = \textcolor{yellow}{\text{Yellow}} \times \textcolor{green}{\text{Green}}\)
2. \(\textcolor{blue}{\text{Blue}} \times \textcolor{yellow}{\text{Yellow}} = 36\)
3. \(\textcolor{green}{\text{Green}} \times \textcolor{pink}{\text{Pink}} = -330\)

### Step-by-Step Solution:

1. **Identify Variables:**
   Let:
   - \(\textcolor{blue}{\text{Blue}} = x\)
   - \(\textcolor{yellow}{\text{Yellow}} = y\)
   - \(\textcolor{red}{\text{Red}} = r\)
   - \(\textcolor{green}{\text{Green}} = g\)
   - \(\textcolor{pink}{\text{Pink}} = p\)

2. **Translate the Equations:**
   - From equation (1): \( r \times g = y \times g \).  
     Simplifying this, if \( g \neq 0 \), then \( r = y \).
   - From equation (2): \( x \times y = 36 \).
   - From equation (3): \( g \times p = -330 \).

3. **Determine Possible Values:**
   - From \( x \times y = 36 \), the factor pairs of 36 are:
     \[
     (1, 36), (-1, -36), (2, 18), (-2, -18), (3, 12), (-3, -12), (4, 9), (-4, -9), (6, 6), (-6, -6)
     \]
   - From \( g \times p = -330 \), the factor pairs of -330 are:
     \[
     (1, -330), (-1, 330), (2, -165), (-2, 165), (3, -110), (-3, 110), (5, -66), (-5, 66), (6, -55), (-6, 55), (10, -33), (-10, 33), (11, -30), (-11, 30), (15, -22), (-15, 22)
     \]

4. **Match Pairs to the Code Pattern:**
   - The code values correspond to the letters based on the alternating positive/negative pattern.
   - For instance, if \(x = 6\), then it could correspond to letter \(F\).
   - Continue finding valid matches for each variable that satisfies the equations and the alternating pattern.

Would you like further details on matching these numbers to the letters or have any other questions?

### Follow-up Questions:
1. What are the exact values of the variables \(\textcolor{blue}{\text{Blue}}, \textcolor{yellow}{\text{Yellow}}, \textcolor{red}{\text{Red}}, \textcolor{green}{\text{Green}},\) and \(\textcolor{pink}{\text{Pink}}\)?
2. How do the alternating positive and negative values influence the choice of letters?
3. What strategies can be used to systematically try each combination of factors?
4. How can you verify that a chosen solution correctly matches all given equations?
5. What is the significance of matching these values correctly to decrypt the word?

**Tip:** When dealing with equations involving variables with unknown assignments, list out potential matches systematically to avoid missing solutions!

Décrypter: Mot caché, chaque lettre de l'alphabet a été codée avec une règle où 1 → A, -2 → B, 3 → C, etc. Les indices incluent: Red × Green = Yellow × Green, Blue × Yellow = 36, Green × Pink = -330. Find the hidden word using these equations.

This page explores how to decrypt a hidden word using algebraic equations and a coding system where letters are assigned alternating positive and negative values. Learn how to solve systems of equations with multiplication properties, appropriate for high school students.

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