Let's determine if Ming has described a proportional relationship by examining the given points in the table.
Firstly, let's list down the given points:
- (5, 10)
- (10, 20)
- (15, 30)
A relationship is considered proportional if the change in [tex]\( y \)[/tex] is consistently a multiple of the change in [tex]\( x \)[/tex], and the line passes through the origin (0,0), essentially meaning [tex]\( y = kx \)[/tex] for some constant [tex]\( k \)[/tex].
1. Checking for Linear Relationship:
- For the point (5, 10): If [tex]\( k \)[/tex] is constant, then [tex]\( y = kx \)[/tex]. Here, [tex]\( 10 = k \cdot 5 \)[/tex], solving this gives [tex]\( k = 2 \)[/tex].
- For the point (10, 20): Here, [tex]\( 20 = k \cdot 10 \)[/tex], solving this gives [tex]\( k = 2 \)[/tex].
- For the point (15, 30): Here, [tex]\( 30 = k \cdot 15 \)[/tex], solving this gives [tex]\( k = 2 \)[/tex].
Since [tex]\( k \)[/tex] is consistently 2 for all points, the relationship is linear, expressed as [tex]\( y = 2x \)[/tex].
2. Checking for Proportional Relationship:
- A proportional relationship must pass through the origin (0,0), meaning when [tex]\( x = 0 \)[/tex], [tex]\( y \)[/tex] should also be 0, adhering to the form [tex]\( y = kx \)[/tex].
Here, the relationship [tex]\( y = 2x \)[/tex] would pass through the origin (since when [tex]\( x = 0 \)[/tex], then [tex]\( y = 2 \cdot 0 = 0 \)[/tex]).
Since the line formed by the points (5, 10), (10, 20), and (15, 30) is both linear and passes through the origin, Ming has, indeed, described a proportional relationship.
Therefore, the correct explanation is:
Ming has described a proportional relationship because the ordered pairs are linear and the line passes through the origin.
