To determine the value of [tex]\( y \)[/tex] when [tex]\( x = 7 \)[/tex] given that [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], follow the detailed steps below:
### Step 1: Identify the Relationship Between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
Since [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], the relationship can be written as [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of proportionality.
### Step 2: Determine the Constant of Proportionality
From the provided table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 2 & 4 & 6 & 10 \\
\hline
y & 12 & 24 & 36 & 60 \\
\hline
\end{array}
\][/tex]
Choose any pair [tex]\((x, y)\)[/tex] from the table to calculate [tex]\( k \)[/tex]. Using the first pair:
[tex]\[
x = 2, \quad y = 12
\][/tex]
Substitute these values into the equation [tex]\( y = kx \)[/tex]:
[tex]\[
12 = k \cdot 2
\][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{12}{2} = 6
\][/tex]
Thus, the constant of proportionality [tex]\( k \)[/tex] is 6.
### Step 3: Calculate [tex]\( y \)[/tex] When [tex]\( x = 7 \)[/tex]
Now, use the constant [tex]\( k \)[/tex] to find [tex]\( y \)[/tex] when [tex]\( x = 7 \)[/tex]:
[tex]\[
y = kx
\][/tex]
Substitute [tex]\( k = 6 \)[/tex] and [tex]\( x = 7 \)[/tex]:
[tex]\[
y = 6 \cdot 7 = 42
\][/tex]
### Step 4: Verify the Answer with Given Choices
Given the possible choices: 37, 42, 48, and 54.
The value of [tex]\( y \)[/tex] when [tex]\( x = 7 \)[/tex] is [tex]\(\boxed{42}\)[/tex].
