To tackle the given problem that [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], we can use the concept of direct variation. When two variables vary directly, their relationship can be described by the equation:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
1. Find the constant [tex]\( k \)[/tex]:
We are given that [tex]\( y = 42 \)[/tex] when [tex]\( x = 7 \)[/tex]. Plugging these values into the direct variation equation:
[tex]\[ 42 = k \cdot 7 \][/tex]
To solve for [tex]\( k \)[/tex], divide both sides of the equation by 7:
[tex]\[ k = \frac{42}{7} = 6.0 \][/tex]
So, the constant of proportionality [tex]\( k \)[/tex] is 6.0.
2. Determine the value of [tex]\( x \)[/tex] when [tex]\( y = 18 \)[/tex]:
Now, we know that [tex]\( y = 18 \)[/tex] and we need to find the corresponding [tex]\( x \)[/tex]. Using the direct variation equation again:
[tex]\[ 18 = 6.0 \cdot x \][/tex]
To solve for [tex]\( x \)[/tex], divide both sides of the equation by 6.0:
[tex]\[ x = \frac{18}{6.0} = 3.0 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] when [tex]\( y \)[/tex] is 18 is:
[tex]\[ x = 3.0 \][/tex]
