Let's analyze the table of points and determine if Ming has described a proportional relationship.
First, let's list the points provided in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
5 & 10 \\
\hline
10 & 20 \\
\hline
15 & 30 \\
\hline
\end{array}
\][/tex]
For the relationship to be proportional, the ratio [tex]\(\frac{y}{x}\)[/tex] must be constant for all points, and the relationship should pass through the origin (0,0).
1. Check the Ratios:
- For the point (5, 10): [tex]\(\frac{10}{5} = 2\)[/tex]
- For the point (10, 20): [tex]\(\frac{20}{10} = 2\)[/tex]
- For the point (15, 30): [tex]\(\frac{30}{15} = 2\)[/tex]
Since the ratios are consistent (always equal to 2), the points form a consistent linear relationship.
2. Check if the Line Passes Through the Origin:
A proportional relationship must pass through the point (0,0). The constant ratio [tex]\(\frac{y}{x} = 2\)[/tex] implies the equation of the line is [tex]\(y = 2x\)[/tex].
Clearly, when [tex]\(x = 0\)[/tex], [tex]\(y = 2 \times 0 = 0\)[/tex]. Thus, the line [tex]\(y = 2x\)[/tex] does pass through the origin.
Based on this information, we can conclude that Ming has indeed described a proportional relationship because:
- The ratio [tex]\(\frac{y}{x}\)[/tex] is constant for all given points.
- The corresponding linear equation passes through the origin.
Hence, the correct explanation is:
"Ming has described a proportional relationship because the ordered pairs are linear and the line pass the origin."
