To determine the value of [tex]\( x \)[/tex] when [tex]\( y \)[/tex] is 30, given that [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], we follow these steps:
1. Understand Direct Variation:
When [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], it means there is a constant ratio [tex]\( k \)[/tex] between [tex]\( y \)[/tex] and [tex]\( x \)[/tex]. This relationship can be written as:
[tex]\[
y = kx
\][/tex]
2. Find the Constant of Proportionality:
We are given that [tex]\( y \)[/tex] is 10 when [tex]\( x \)[/tex] is 8. Using these values, we can find the constant [tex]\( k \)[/tex].
[tex]\[
k = \frac{y}{x} = \frac{10}{8} = 1.25
\][/tex]
3. Set Up the Equation with the Found Constant:
To find [tex]\( x \)[/tex] when [tex]\( y \)[/tex] is 30, we use the same relationship [tex]\( y = kx \)[/tex]. We know that:
[tex]\[
y = 30
\][/tex]
and we have already determined that [tex]\( k = 1.25 \)[/tex]. So we have:
[tex]\[
30 = 1.25x
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], solve the equation for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{30}{1.25} = 24
\][/tex]
Therefore, when [tex]\( y \)[/tex] is 30, [tex]\( x \)[/tex] is [tex]\( 24 \)[/tex].
So, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = 24
\][/tex]
