To find the missing number, denoted as `?`, let's carefully analyze the patterns in the table:
[tex]\[
\begin{tabular}{rrrr}
-4 & -6 & -9 & -6 \\
? & 7 & 4 & -4 \\
9 & 3 & 2 & 6 \\
3 & -4 & 8 & -6
\end{tabular}
\][/tex]
Given the problem mentions that there is an algebraic equation involving subtraction, multiplication, and division that repeats across all rows, we need to identify this pattern to determine the value of `?`.
1. Examine Vertical Relationships:
- Look for patterns in each column of the table.
Column 1:
- From 3 to 9: [tex]\( 9 - 3 = 6 \)[/tex]
Column 2:
- From -4 to 3: [tex]\( 3 - (-4) = 3 + 4 = 7 \)[/tex]
Column 4:
- From -6 to 6: [tex]\( 6 - (-6) = 6 + 6 = 12 \)[/tex]
2. Determine if a Single Transformation Pattern Exists:
- From the analysis, the transformations for Columns 1, 2, and 4 show a clear pattern of addition.
3. Apply the Pattern:
- Using the identified patterns, we can apply them to deduce the missing value `?`.
4. Solve Using Vertical Transformation:
- We see that if we go from 3 to 9 (in Column 1), the pattern is to add 6, and similarly in other columns, we are adding specific values.
- Notice the numeric jump between specific entries to find the missing value:
If we add 7 to some value to get the number in the row underneath, we get the missing value `?` for the first column:
Let's denote the missing value `?` in the first column of the second row (highlighted):
[tex]\[
? + 7 = -4
\][/tex]
So, solving for `?`:
[tex]\[
? = -4 - 7 = -11
\][/tex]
Therefore, the number that completes the pattern is:
[tex]\[
\boxed{-11}
\][/tex]
