Given that [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], we can express this relationship with the equation:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
To find [tex]\( k \)[/tex], we use the given values: [tex]\( y = 32 \)[/tex] when [tex]\( x = 8 \)[/tex].
Substitute these values into the equation:
[tex]\[ 32 = k \cdot 8 \][/tex]
Solving for [tex]\( k \)[/tex], we get:
[tex]\[ k = \frac{32}{8} = 4 \][/tex]
Now, we have the proportionality constant [tex]\( k = 4 \)[/tex].
Next, we need to find the value of [tex]\( x \)[/tex] when [tex]\( y \)[/tex] is 12.
Using the direct variation equation [tex]\( y = kx \)[/tex] and substituting [tex]\( y = 12 \)[/tex] and [tex]\( k = 4 \)[/tex]:
[tex]\[ 12 = 4x \][/tex]
Solving for [tex]\( x \)[/tex], we divide both sides of the equation by 4:
[tex]\[ x = \frac{12}{4} = 3 \][/tex]
Therefore, when [tex]\( y \)[/tex] is 12, [tex]\( x \)[/tex] is:
[tex]\[ x = 3 \][/tex]
