Given that [tex]$y$[/tex] varies directly with [tex]$x$[/tex] in the table below, what is the value of [tex]$y$[/tex] if the value of [tex]$x$[/tex] is 7?

\begin{tabular}{|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & 2 & 4 & 6 & 10 \\
\hline
[tex]$y$[/tex] & 12 & 24 & 36 & 60 \\
\hline
\end{tabular}

A. 37
B. 42
C. 48
D. 54



Answer :

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].