To determine the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 14, given that [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], we can use the information provided. Let's break this down step-by-step.
1. Understanding Direct Variation:
When [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], it means that the ratio [tex]\( \frac{y}{x} \)[/tex] is constant. This relationship can be written as:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is a constant.
2. Finding the Constant [tex]\( k \)[/tex]:
We can find the constant [tex]\( k \)[/tex] using any known pair of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values. We are given:
[tex]\[ x = 2, y = 3 \][/tex]
Substituting these values into the equation [tex]\( y = kx \)[/tex]:
[tex]\[ 3 = k \cdot 2 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{3}{2} \][/tex]
So, [tex]\( k = 1.5 \)[/tex].
3. Verifying the Constant [tex]\( k \)[/tex] Using Another Pair:
To ensure our constant [tex]\( k \)[/tex] is correct, we can use the second pair of values given:
[tex]\[ x = 10, y = 15 \][/tex]
Substituting these values into the equation [tex]\( y = kx \)[/tex]:
[tex]\[ 15 = k \cdot 10 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{15}{10} = 1.5 \][/tex]
This confirms that [tex]\( k = 1.5 \)[/tex] is consistent.
4. Finding [tex]\( y \)[/tex] for [tex]\( x = 14 \)[/tex]:
Now, we use the constant [tex]\( k \)[/tex] to find [tex]\( y \)[/tex] when [tex]\( x = 14 \)[/tex]:
[tex]\[ y = kx \][/tex]
Substituting [tex]\( k = 1.5 \)[/tex] and [tex]\( x = 14 \)[/tex]:
[tex]\[ y = 1.5 \cdot 14 \][/tex]
Calculating the value:
[tex]\[ y = 21.0 \][/tex]
Therefore, when [tex]\( x = 14 \)[/tex], the value of [tex]\( y \)[/tex] is [tex]\( \boxed{21} \)[/tex].
