Certainly! To solve this problem, let's follow a series of logical steps. We know that when two variables vary directly, their ratio remains constant. In this case, we understand that [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex].
1. Determine the constant of variation (k):
- Direct variation means [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of proportionality.
- Given [tex]\( y = 70 \)[/tex] when [tex]\( x = 14 \)[/tex], we can identify [tex]\( k \)[/tex] by substituting these values into the direct variation equation:
[tex]\[
70 = k \cdot 14
\][/tex]
- Solving for [tex]\( k \)[/tex], we divide both sides by 14:
[tex]\[
k = \frac{70}{14} = 5
\][/tex]
2. Use the constant of variation to find the new value of [tex]\( y \)[/tex]:
- Now that we know the constant [tex]\( k \)[/tex], we can use it to find [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex].
- Substitute [tex]\( k = 5 \)[/tex] and [tex]\( x = 3 \)[/tex] into the direct variation equation:
[tex]\[
y = kx = 5 \cdot 3
\][/tex]
- Hence, we find:
[tex]\[
y = 15
\][/tex]
So, when [tex]\( x = 3 \)[/tex], [tex]\( y \)[/tex] is [tex]\( 15 \)[/tex].
