What number completes the pattern? Note that there is a single algebraic equation (limited to the basic arithmetic operations of addition, subtraction, multiplication, and division) that repeats across all rows.

6 -5 -6 5
-4 3 2 -6
6 6 9 4
-9 ? 6 3

To solve this, we need to identify the algebraic equation that connects the numbers in each row. Let's analyze each row to deduce the potential pattern.

### Row 1: \( 6, -5, -6, 5 \)
- \( 6 + (-5) + (-6) + 5 = 0 \)

### Row 2: \( -4, 3, 2, -6 \)
- \( -4 + 3 + 2 + (-6) = -5 \)

### Row 3: \( 6, 6, 9, 4 \)
- \( 6 + 6 + 9 + 4 = 25 \)

### Row 4: \( -9, ?, 6, 3 \)
- The sum of the known numbers is \( -9 + 6 + 3 = 0 \).

Now, let's hypothesize that the pattern involves ensuring the sum of all four numbers in each row equals a fixed value.

- In **Row 1**, the sum is 0.
- In **Row 2**, the sum is -5.
- In **Row 3**, the sum is 25.
- Therefore, in **Row 4**, the sum of all four numbers should be equal to 0 (as in Row 1).

Since the sum of the three known numbers is \( 0 \), the missing number must be such that the total remains unchanged at 0. Thus, the missing number is **0**.

So, the complete Row 4 is:
\[
-9, 0, 6, 3
\]

Would you like more details on how to confirm this, or have any other questions?

### Related Questions:
1. How can we generalize patterns in arithmetic sequences?
2. What types of sequences rely on summing row values?
3. How does algebraic manipulation help in solving number patterns?
4. Can patterns like this exist with non-arithmetic operations?
5. How would introducing multiplication or division affect the patterns?

**Tip**: Always check for both simple and complex operations when deciphering number patterns in rows or sequences.

What number completes the pattern? Note that there is a single algebraic equation (limited to the basic arithmetic operations of addition, subtraction, multiplication, and division) that repeats across all rows. 6 -5 -6 5 -4 3 2 -6 6 6 9 4 -9 ? 6 3

Discover how to find the missing number in an arithmetic sequence involving addition and subtraction. Learn how the sum of row elements follows a fixed pattern in this problem. Suitable for students in Grades 6-8, this step-by-step solution covers core algebra and arithmetic principles.

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