When dealing with direct variation between two variables, [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], which can be expressed mathematically as [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of proportionality.
Given the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
1. [tex]\( x_1 = 2 \)[/tex] and [tex]\( y_1 = 3 \)[/tex]
2. [tex]\( x_2 = 10 \)[/tex] and [tex]\( y_2 = 15 \)[/tex]
First, we need to determine the constant of proportionality [tex]\( k \)[/tex]. We can use any of the pairs [tex]\((x, y)\)[/tex] to find [tex]\( k \)[/tex]. Let's use the first pair to find [tex]\( k \)[/tex]:
[tex]\[ y_1 = kx_1 \][/tex]
[tex]\[ 3 = k \cdot 2 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{3}{2} = 1.5 \][/tex]
Now, we have established that [tex]\( k = 1.5 \)[/tex].
Next, we use this constant of proportionality to find the value of [tex]\( y \)[/tex] when [tex]\( x = 14 \)[/tex]:
[tex]\[ y = kx \][/tex]
[tex]\[ y = 1.5 \cdot 14 \][/tex]
Calculating the value:
[tex]\[ y = 21 \][/tex]
So, the value of [tex]\( y \)[/tex] when [tex]\( x = 14 \)[/tex] is [tex]\( 21 \)[/tex].
