To determine whether Ming has described a proportional relationship with the given points, we need to check if the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is proportional. A proportional relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] can be described by the equation [tex]\(y = kx\)[/tex], where [tex]\(k\)[/tex] is a constant ratio, and the line passes through the origin [tex]\((0,0)\)[/tex].
Let’s analyze the points provided in the table:
[tex]\[
\begin{array}{|r|r|}
\hline
x & y \\
\hline
5 & 10 \\
\hline
10 & 20 \\
\hline
15 & 30 \\
\hline
\end{array}
\][/tex]
1. Check for a constant ratio [tex]\(k\)[/tex]:
For each point [tex]\((x, y)\)[/tex], calculate the ratio [tex]\( \frac{y}{x} \)[/tex]:
[tex]\[
\frac{10}{5} = 2
\][/tex]
[tex]\[
\frac{20}{10} = 2
\][/tex]
[tex]\[
\frac{30}{15} = 2
\][/tex]
The ratio [tex]\( \frac{y}{x} \)[/tex] is the same for all three points, and [tex]\( k = 2 \)[/tex]. This indicates that the points lie on a line described by the equation [tex]\( y = 2x \)[/tex].
2. Check if the line passes through the origin [tex]\((0,0)\)[/tex]:
Substitute [tex]\(x = 0\)[/tex] in the proportional relationship [tex]\( y = 2x \)[/tex]:
[tex]\[
y = 2 \times 0 = 0
\][/tex]
This confirms that the line passes through the origin [tex]\((0,0)\)[/tex].
Since the relationship described by the points has a constant ratio (slope) [tex]\( k = 2 \)[/tex] and the line passes through the origin, Ming has described a proportional relationship.
Thus, the correct explanation is:
Ming has described a proportional relationship because the ordered pairs are linear and the line passes through the origin.
