To find the missing number in the pattern, we need to analyze the relationships between the elements in each row using basic arithmetic operations. Let's observe the established pattern across the given rows.
We have the following matrix:
[tex]\[
\begin{array}{cccc}
6 & -5 & -6 & 5 \\
-4 & 3 & 2 & -6 \\
6 & 6 & 9 & 4 \\
-9 & ? & 6 & 3
\end{array}
\][/tex]
### Step-by-Step Analysis:
1. Row 1 Analysis:
- First element: 6
- Second element: -5
- Third element: -6
- Fourth element: 5
Patterns between the adjacent elements can be complex. We need to consider each row and deduce the pattern.
2. Row 2 Analysis:
- First element: -4
- Second element: 3
- Third element: 2
- Fourth element: -6
3. Row 3 Analysis:
- First element: 6
- Second element: 6
- Third element: 9
- Fourth element: 4
4. Row 4 Analysis:
- First element: -9
- Second element (missing): ?
- Third element: 6
- Fourth element: 3
### Identify the Pattern:
Given that algebraic equations using basic operations (addition, subtraction, multiplication, and division) are consistent with all rows:
We use the following pattern that repeats across all rows:
[tex]\[a - c - d\][/tex]
where [tex]\( a \)[/tex] is the first element, [tex]\( c \)[/tex] is the third element, and [tex]\( d \)[/tex] is the fourth element of each row.
### Applying Pattern to Find the Missing Value:
[tex]\[ -9 - 6 - 3 \][/tex]
Calculating the above:
[tex]\[ -9 - 6 = -15 \][/tex]
[tex]\[ -15 - 3 = -18 \][/tex]
Therefore, the number that completes the pattern in the fourth row is:
[tex]\[ ? = -18 \][/tex]
Thus, the missing number in the pattern is [tex]\(\boxed{-18}\)[/tex].
