Answer :
To determine in which table [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], we must check for a consistent ratio [tex]\( \frac{y}{x} \)[/tex]. That is, if [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], then [tex]\( \frac{y}{x} \)[/tex] should yield the same constant value for all pairs [tex]\((x, y)\)[/tex] in the table.
Let's analyze each table:
Table A:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -2 \\ 2 & -4 \\ 3 & -16 \\ \hline \end{array} \][/tex]
Calculating the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
[tex]\[ \frac{-2}{1} = -2, \quad \frac{-4}{2} = -2, \quad \frac{-16}{3} \approx -5.33 \][/tex]
The ratio [tex]\( \frac{y}{x} \)[/tex] is not consistent, so [tex]\( y \)[/tex] does not vary directly with [tex]\( x \)[/tex] in Table A.
Table B:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -5 \\ 2 & 18 \\ 3 & 41 \\ \hline \end{array} \][/tex]
Calculating the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
[tex]\[ \frac{-5}{1} = -5, \quad \frac{18}{2} = 9, \quad \frac{41}{3} \approx 13.67 \][/tex]
The ratio [tex]\( \frac{y}{x} \)[/tex] is not consistent, so [tex]\( y \)[/tex] does not vary directly with [tex]\( x \)[/tex] in Table B.
Table C:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 26 \\ 2 & 52 \\ 3 & 78 \\ \hline \end{array} \][/tex]
Calculating the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
[tex]\[ \frac{26}{1} = 26, \quad \frac{52}{2} = 26, \quad \frac{78}{3} = 26 \][/tex]
The ratio [tex]\( \frac{y}{x} \)[/tex] is consistent at 26 for all pairs. Therefore, [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] in Table C.
Table D:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -7 \\ 2 & -1 \\ 3 & 6 \\ \hline \end{array} \][/tex]
Calculating the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
[tex]\[ \frac{-7}{1} = -7, \quad \frac{-1}{2} = -0.5, \quad \frac{6}{3} = 2 \][/tex]
The ratio [tex]\( \frac{y}{x} \)[/tex] is not consistent, so [tex]\( y \)[/tex] does not vary directly with [tex]\( x \)[/tex] in Table D.
In summary, the correct table in which [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] is:
Table C:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 26 \\ 2 & 52 \\ 3 & 78 \\ \hline \end{array} \][/tex]
Therefore, the correct answer is [tex]\( \boxed{3} \)[/tex].
Let's analyze each table:
Table A:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -2 \\ 2 & -4 \\ 3 & -16 \\ \hline \end{array} \][/tex]
Calculating the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
[tex]\[ \frac{-2}{1} = -2, \quad \frac{-4}{2} = -2, \quad \frac{-16}{3} \approx -5.33 \][/tex]
The ratio [tex]\( \frac{y}{x} \)[/tex] is not consistent, so [tex]\( y \)[/tex] does not vary directly with [tex]\( x \)[/tex] in Table A.
Table B:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -5 \\ 2 & 18 \\ 3 & 41 \\ \hline \end{array} \][/tex]
Calculating the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
[tex]\[ \frac{-5}{1} = -5, \quad \frac{18}{2} = 9, \quad \frac{41}{3} \approx 13.67 \][/tex]
The ratio [tex]\( \frac{y}{x} \)[/tex] is not consistent, so [tex]\( y \)[/tex] does not vary directly with [tex]\( x \)[/tex] in Table B.
Table C:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 26 \\ 2 & 52 \\ 3 & 78 \\ \hline \end{array} \][/tex]
Calculating the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
[tex]\[ \frac{26}{1} = 26, \quad \frac{52}{2} = 26, \quad \frac{78}{3} = 26 \][/tex]
The ratio [tex]\( \frac{y}{x} \)[/tex] is consistent at 26 for all pairs. Therefore, [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] in Table C.
Table D:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -7 \\ 2 & -1 \\ 3 & 6 \\ \hline \end{array} \][/tex]
Calculating the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
[tex]\[ \frac{-7}{1} = -7, \quad \frac{-1}{2} = -0.5, \quad \frac{6}{3} = 2 \][/tex]
The ratio [tex]\( \frac{y}{x} \)[/tex] is not consistent, so [tex]\( y \)[/tex] does not vary directly with [tex]\( x \)[/tex] in Table D.
In summary, the correct table in which [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] is:
Table C:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 26 \\ 2 & 52 \\ 3 & 78 \\ \hline \end{array} \][/tex]
Therefore, the correct answer is [tex]\( \boxed{3} \)[/tex].
