To solve this problem, we need to understand the concept of direct variation. If [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], it means that [tex]\( y \)[/tex] can be expressed as [tex]\( y = kx \)[/tex] for some constant [tex]\( k \)[/tex].
Given:
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 7 \)[/tex].
First, we need to find the constant of variation [tex]\( k \)[/tex]. We can do this by using the provided values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ y = kx \][/tex]
Substitute [tex]\( y = 7 \)[/tex] and [tex]\( x = 5 \)[/tex]:
[tex]\[ 7 = k \cdot 5 \][/tex]
To find [tex]\( k \)[/tex], solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{7}{5} \][/tex]
Now that we have the constant of variation [tex]\( k \)[/tex], we can use it to find the value of [tex]\( y \)[/tex] when [tex]\( x = 10 \)[/tex]. Using the equation [tex]\( y = kx \)[/tex], substitute [tex]\( k \)[/tex] and the new value of [tex]\( x \)[/tex]:
[tex]\[ y = \frac{7}{5} \cdot 10 \][/tex]
To solve this:
[tex]\[ y = 1.4 \cdot 10 \][/tex]
[tex]\[ y = 14 \][/tex]
Therefore, when [tex]\( x = 10 \)[/tex], [tex]\( y \)[/tex] is [tex]\( 14 \)[/tex].
