Certainly! To find the value of [tex]\( x \)[/tex] when [tex]\( y \)[/tex] is 18 given that [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], follow these steps:
1. Determine the constant of proportionality [tex]\( k \)[/tex]:
Since [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], we can express this relationship as [tex]\( y = kx \)[/tex] where [tex]\( k \)[/tex] is the constant of proportionality.
From the given information, [tex]\( y = 42 \)[/tex] when [tex]\( x = 7 \)[/tex]. Substituting these values into the equation [tex]\( y = kx \)[/tex]:
[tex]\[
42 = k \cdot 7
\][/tex]
To find [tex]\( k \)[/tex], solve for [tex]\( k \)[/tex] by dividing both sides of the equation by 7:
[tex]\[
k = \frac{42}{7} = 6
\][/tex]
So, the constant of proportionality [tex]\( k \)[/tex] is 6.
2. Use the constant of proportionality to find [tex]\( x \)[/tex] when [tex]\( y \)[/tex] is 18:
Now that we know [tex]\( k = 6 \)[/tex], we can use this to find [tex]\( x \)[/tex] when [tex]\( y \)[/tex] is 18.
Using the direct variation formula [tex]\( y = kx \)[/tex], substitute [tex]\( y = 18 \)[/tex] and [tex]\( k = 6 \)[/tex]:
[tex]\[
18 = 6 \cdot x
\][/tex]
To find [tex]\( x \)[/tex], isolate [tex]\( x \)[/tex] by dividing both sides of the equation by 6:
[tex]\[
x = \frac{18}{6} = 3
\][/tex]
Therefore, when [tex]\( y \)[/tex] is 18, the corresponding value of [tex]\( x \)[/tex] is 3.
So the answer is:
[tex]\[
x = 3
\][/tex]
