To determine which table contains only points that lie on the line given by the equation [tex]\( y = 6x - 6 \)[/tex], we need to check each pair of [tex]\((x, y)\)[/tex] values in the tables and see if they satisfy the equation.
### Table F
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & -6 \\
\hline
5 & 5 \\
\hline
\end{array}
\][/tex]
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 6(0) - 6 = -6
\][/tex]
The point [tex]\((0, -6)\)[/tex] satisfies the equation.
2. For [tex]\( x = 5 \)[/tex]:
[tex]\[
y = 6(5) - 6 = 30 - 6 = 24
\][/tex]
The point [tex]\((5, 5)\)[/tex] does not satisfy the equation since [tex]\( y \)[/tex] should be 24 instead of 5.
Hence, Table F does not contain only points that lie on the line.
### Table G
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 6 \\
\hline
-2 & -18 \\
\hline
\end{array}
\][/tex]
1. For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 6(2) - 6 = 12 - 6 = 6
\][/tex]
The point [tex]\((2, 6)\)[/tex] satisfies the equation.
2. For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = 6(-2) - 6 = -12 - 6 = -18
\][/tex]
The point [tex]\(( -2, -18)\)[/tex] satisfies the equation.
Hence, Table G contains only points that lie on the line.
### Table H
[tex]\[
\begin{array}{|r|l|}
\hline
x & y \\
\hline
2 & 6 \\
\hline
-1 & 0 \\
\hline
\end{array}
\][/tex]
1. For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 6(2) - 6 = 12 - 6 = 6
\][/tex]
The point [tex]\((2, 6)\)[/tex] satisfies the equation.
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = 6(-1) - 6 = -6 - 6 = -12
\][/tex]
The point [tex]\(( -1, 0)\)[/tex] does not satisfy the equation since [tex]\( y \)[/tex] should be -12 instead of 0.
Hence, Table H does not contain only points that lie on the line.
### Table J
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 6 \\
\hline
3 & 12 \\
\hline
\end{array}
\][/tex]
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 6(0) - 6 = -6
\][/tex]
The point [tex]\((0, 6)\)[/tex] does not satisfy the equation since [tex]\( y \)[/tex] should be -6 instead of 6.
2. For [tex]\( x = 3 \)[/tex]:
[tex]\[
y = 6(3) - 6 = 18 - 6 = 12
\][/tex]
The point [tex]\((3, 12)\)[/tex] satisfies the equation.
Hence, Table J does not contain only points that lie on the line.
After checking all the tables, we find that only Table G contains points that both lie on the line described by the equation [tex]\( y = 6x - 6 \)[/tex].
