To find the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 3, given the relationship that [tex]\( y \)[/tex] varies inversely with [tex]\( x \)[/tex], we can follow these steps:
1. Understand the inverse variation: When [tex]\( y \)[/tex] varies inversely with [tex]\( x \)[/tex], it means [tex]\( y = \frac{k}{x} \)[/tex], where [tex]\( k \)[/tex] is a constant. This relationship means that the product of [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is always the same constant value [tex]\( k \)[/tex].
2. Determine the constant [tex]\( k \)[/tex]: We are given that [tex]\( y = 3 \)[/tex] when [tex]\( x = 11 \)[/tex]. Using the inverse variation formula, we can set up the equation:
[tex]\[
3 = \frac{k}{11}
\][/tex]
We solve for [tex]\( k \)[/tex] by multiplying both sides of the equation by 11:
[tex]\[
k = 3 \times 11 = 33
\][/tex]
So, the constant [tex]\( k \)[/tex] is 33.
3. Find the new value of [tex]\( y \)[/tex]: Now we need to find the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 3. Using the inverse variation formula again with the calculated constant [tex]\( k \)[/tex]:
[tex]\[
y = \frac{33}{3}
\][/tex]
Dividing 33 by 3, we get:
[tex]\[
y = 11
\][/tex]
Therefore, when [tex]\( x \)[/tex] is 3, the value of [tex]\( y \)[/tex] is [tex]\( 11 \)[/tex].
