To determine whether the relationships given in the table have the same constant of proportionality, we will evaluate the constant for each pair of [tex]\((x, y)\)[/tex] coordinates. The constant of proportionality ([tex]\(k\)[/tex]) can be found using the formula:
[tex]\[ k = \frac{y}{x} \][/tex]
Let's do this step by step:
### Step 1: Calculate the constant of proportionality for the first pair.
Given the coordinates [tex]\((x_1, y_1) = (2, 7)\)[/tex],
[tex]\[ k_1 = \frac{y_1}{x_1} = \frac{7}{2} = 3.5 \][/tex]
### Step 2: Calculate the constant of proportionality for the second pair.
Given the coordinates [tex]\((x_2, y_2) = (9, 31.5)\)[/tex],
[tex]\[ k_2 = \frac{y_2}{x_2} = \frac{31.5}{9} = 3.5 \][/tex]
### Step 3: Compare the constants of proportionality.
From our calculations:
[tex]\[ k_1 = 3.5 \][/tex]
[tex]\[ k_2 = 3.5 \][/tex]
Since both constants of proportionality are equal ([tex]\(k_1 = k_2 = 3.5\)[/tex]), we can conclude that both relationships have the same constant of proportionality.
Therefore, the pair of values given in the table [tex]\((2, 7)\)[/tex] and [tex]\((9, 31.5)\)[/tex] both follow relationships where [tex]\(y\)[/tex] is directly proportional to [tex]\(x\)[/tex] with a constant of proportionality equal to 3.5.
