Let's evaluate the viability of the solutions given in both tables provided.
Recall the relationship between the number of books [tex]\( b \)[/tex] and their total weight [tex]\( w \)[/tex]:
[tex]\[ w = 6b \][/tex]
We will use this relationship to check if each pair [tex]\((b, w)\)[/tex] satisfies the equation [tex]\( w = 6b \)[/tex].
### Table 1
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Books } (b) & \text{Weight } (w) \\
\hline
-2 & -12 \\
-1 & -6 \\
0 & 0 \\
1 & 6 \\
2 & 12 \\
\hline
\end{array}
\][/tex]
Let's check each pair one by one:
- For [tex]\( b = -2 \)[/tex]:
[tex]\[ w = 6(-2) = -12 \][/tex]
The pair [tex]\((-2, -12)\)[/tex] is valid.
- For [tex]\( b = -1 \)[/tex]:
[tex]\[ w = 6(-1) = -6 \][/tex]
The pair [tex]\((-1, -6)\)[/tex] is valid.
- For [tex]\( b = 0 \)[/tex]:
[tex]\[ w = 6(0) = 0 \][/tex]
The pair [tex]\( (0, 0) \)[/tex] is valid.
- For [tex]\( b = 1 \)[/tex]:
[tex]\[ w = 6(1) = 6 \][/tex]
The pair [tex]\( (1, 6) \)[/tex] is valid.
- For [tex]\( b = 2 \)[/tex]:
[tex]\[ w = 6(2) = 12 \][/tex]
The pair [tex]\( (2, 12) \)[/tex] is valid.
### Table 2
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Books } (b) & \text{Weight } (w) \\
\hline
-1 & -6 \\
-0.5 & -3 \\
0 & 0 \\
0.5 & 3 \\
1 & 6 \\
\hline
\end{array}
\][/tex]
Let's check each pair one by one:
- For [tex]\( b = -1 \)[/tex]:
[tex]\[ w = 6(-1) = -6 \][/tex]
The pair [tex]\((-1, -6)\)[/tex] is valid.
- For [tex]\( b = -0.5 \)[/tex]:
[tex]\[ w = 6(-0.5) = -3 \][/tex]
The pair [tex]\((-0.5, -3)\)[/tex] is valid.
- For [tex]\( b = 0 \)[/tex]:
[tex]\[ w = 6(0) = 0 \][/tex]
The pair [tex]\( (0, 0) \)[/tex] is valid.
- For [tex]\( b = 0.5 \)[/tex]:
[tex]\[ w = 6(0.5) = 3 \][/tex]
The pair [tex]\((0.5, 3)\)[/tex] is valid.
- For [tex]\( b = 1 \)[/tex]:
[tex]\[ w = 6(1) = 6 \][/tex]
The pair [tex]\( (1, 6) \)[/tex] is valid.
After verifying both tables, it is clear that all pairs in both tables satisfy the relationship [tex]\( w = 6b \)[/tex].
### Conclusion
Both tables contain only viable solutions where [tex]\( w = 6b \)[/tex].
Thus, both tables are viable.
