To determine whether Ming has described a proportional relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex], let's follow these steps:
1. Identify the given points: We have the pairs [tex]\((5, 10)\)[/tex], [tex]\((10, 20)\)[/tex], and [tex]\((15, 30)\)[/tex].
2. Calculate the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
- For the point [tex]\((5, 10)\)[/tex]: [tex]\( \frac{10}{5} = 2 \)[/tex]
- For the point [tex]\((10, 20)\)[/tex]: [tex]\( \frac{20}{10} = 2 \)[/tex]
- For the point [tex]\((15, 30)\)[/tex]: [tex]\( \frac{30}{15} = 2 \)[/tex]
3. Check for consistency in the ratios:
- Since the ratio [tex]\( \frac{y}{x} = 2 \)[/tex] is the same for all given points, this indicates a proportional relationship.
4. Verify if the relationship passes through the origin:
- In a proportional relationship, the line described by the pairs must pass through the origin [tex]\((0, 0)\)[/tex].
- The proportional constant ratio [tex]\( \frac{y}{x} = 2 \)[/tex] implies that [tex]\( y = 2x \)[/tex].
Therefore, each of these points fits the equation [tex]\(y = 2x \)[/tex], where if [tex]\(x = 0\)[/tex], then [tex]\(y = 2 \cdot 0 = 0\)[/tex], confirming that the line passes through the origin.
Based on these calculations, Ming has indeed described a proportional relationship because the ratios are consistent across all points, and the line passes through the origin.
Thus, the correct explanation is:
Ming has described a proportional relationship because the ordered pairs are linear and the line passes through the origin.
