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

[tex]\[
\begin{tabular}{cccc}
-4 & -6 & -9 & -6 \\
? & 7 & 4 & -4 \\
9 & 3 & 2 & 6 \\
3 & -4 & 8 & -6
\end{tabular}
\][/tex]



Answer :

Let's determine the pattern by closely examining the given numbers and logical relationships from one or more rows.

Recall the row and notice such equations, which can be derived logically:

1. For the first row: [tex]\[-4 + -6 \times b = -9 + -6\][/tex]

2. For the second row: [tex]\[a + 7 \times b = 4 + -4\][/tex]

3. For the third row: [tex]\[9 + 3 \times c = 2 + 6\][/tex]

4. For the fourth row: [tex]\[3 + -4 \times d = 8 + -6\][/tex]

Having these equations above, solving further for [tex]\(a\)[/tex]:

### Solve for [tex]\(b\)[/tex]:
From row 1:
[tex]\[ -4 + -6 \times b = -9 + -6 \][/tex]
Simplify the equation:
[tex]\[ -4 - 6b = -15 \][/tex]
Add 4 to both sides:
[tex]\[ -6b = -11 \][/tex]
Divide by -6:
[tex]\[ b = \frac{11}{6} \][/tex]

### Solve for [tex]\(c\)[/tex]:
From row 3:
[tex]\[ 9 + 3 \times c = 2 + 6 \][/tex]
Simplify the equation:
[tex]\[ 9 + 3c = 8 \][/tex]
Subtract 9 from both sides:
[tex]\[ 3c = -1 \][/tex]
Divide by 3:
[tex]\[ c = -\frac{1}{3} \][/tex]

### Solve for [tex]\(d\)[/tex]:
From row 4:
[tex]\[ 3 + -4 \times d = 8 + -6 \][/tex]
Simplify the equation:
[tex]\[ 3 - 4d = 2 \][/tex]
Subtract 3 from both sides:
[tex]\[ -4d = -1 \][/tex]
Divide by -4:
[tex]\[ d = \frac{1}{4} \][/tex]

### Solve for [tex]\(a\)[/tex] while knowing [tex]\(b = \frac{11}{6}\)[/tex]:
From row 2:
[tex]\[ a + 7 \times b = 0 \][/tex]
Substitute [tex]\(b = \frac{11}{6}\)[/tex] in the equation,
[tex]\[ a + 7 \times \frac{11}{6} = 0 \][/tex]
Solve the equation:
[tex]\[ a + \frac{77}{6} = 0 \][/tex]
Subtract [tex]\(\frac{77}{6}\)[/tex] from both sides:
[tex]\[ a = -\frac{77}{6} \][/tex]

Thus, the missing value for [tex]\(a\)[/tex] which completes the pattern is:
[tex]\[ a = -\frac{77}{6} \][/tex]