To identify the number that completes the pattern, we need to find the underlying algebraic equation that applies to each row. Observing the given numbers, we can start by analyzing the pattern in each row.

Let's denote the numbers in each row as follows:

• First row: $a_1 = 6$, $a_2 = -5$, $a_3 = -6$, $a_4 = 5$
• Second row: $b_1 = -4$, $b_2 = 3$, $b_3 = 2$, $b_4 = -6$
• Third row: $c_1 = 6$, $c_2 = 6$, $c_3 = 9$, $c_4 = 4$
• Fourth row: $d_1 = -9$, $d_2 = ?$, $d_3 = 6$, $d_4 = 3$

By checking each row, we can try to establish a pattern or relationship:

First Row:

Let's check the differences between consecutive numbers: $6 - (-5) = 11$ $-5 - (-6) = 1$ $-6 - 5 = -11$

The differences are $11, 1, -11$.

Second Row:

$-4 - 3 = -7$ $3 - 2 = 1$ $2 - (-6) = 8$

The differences are $-7, 1, 8$.

Third Row:

$6 - 6 = 0$ $6 - 9 = -3$ $9 - 4 = 5$

The differences are $0, -3, 5$.

Fourth Row:

$-9 - ?$ $? - 6$ $6 - 3 = 3$

Assuming the second difference ($? - 6$) is consistent with the previous rows' second difference, which is $1$, we have: $? - 6 = 1$ $? = 7$

Next, verify the algebraic equation based on the first difference in the fourth row: $-9 - 7 = -16$

There is an inconsistency in establishing a single arithmetic equation, given the unique nature of the numbers and differences.

Instead, a common approach might be the sum of the terms in each row should match or follow a specific pattern. Let's check the sum of the first three rows: $6 + (-5) + (-6) + 5 = 0$ $-4 + 3 + 2 + (-6) = -5$ $6 + 6 + 9 + 4 = 25$

Given no common arithmetic pattern, we apply the differences approach established earlier. Recheck for validation: $-9 - ? + 6 + 3 = -7 -1 - (-11) = -7$

Thus, the most plausible value by process of elimination is: $7$

Therefore, the number that completes the pattern is $7$.

Do you want more details or have any questions?

Here are 5 numbered questions you might ask:

1. Can you explain the steps to solve a specific type of algebraic equation?
2. How do you find the common difference in an arithmetic sequence?
3. What are some tips for identifying patterns in number sequences?
4. Can you help solve a calculus problem involving derivatives?
5. How do you approach solving complex word problems in math?

Tip: Always double-check your calculations by substituting the found value back into the sequence to ensure consistency with the pattern.