To determine which of the given tables represents a linear relationship that is also proportional, we need to look for two key characteristics:
- A linear relationship means the relationship between \(x\) and \(y\) can be expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- Proportional relationships are a special case of linear relationships where \(b = 0\) (i.e., they pass through the origin, and can be expressed in the form \(y = mx\)).
Let's analyze each table:
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Table 1: \[ \begin{array}{|c|c|} \hline x & y \ \hline 0 & 4 \ 2 & 6 \ 4 & 8 \ \hline \end{array} \]
- When \(x = 0\), \(y = 4\) (not proportional).
- This relationship is linear, but not proportional because it does not pass through the origin.
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Table 2: \[ \begin{array}{|c|c|} \hline x & y \ \hline 0 & 0 \ 6 & 3 \ 12 & 6 \ \hline \end{array} \]
- When \(x = 0\), \(y = 0\) (this passes through the origin).
- To check for proportionality, we can calculate the ratio \( y/x \):
- For \(x=6\), \(y=3\) → \( \frac{3}{6} = \frac{1}{2} \).
- For \(x=12\), \(y=6\) → \( \frac{6}{12} = \frac{1}{2} \).
- The ratio is constant (1/2), so this represents a proportional relationship and is linear.
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Table 3: \[ \begin{array}{|c|c|} \hline x & y \ \hline 0 & 3 \ 5 & 5 \ 10 & 7 \ \hline \end{array} \]
- When \(x = 0\), \(y = 3\) (not proportional).
- This relationship is not proportional because it does not pass through the origin. Although it might show a linear trend, it does not meet proportionality.
Conclusion: The table that represents a linear relationship that is also proportional is Table 2.