Certainly! Let's solve the problem step by step.
First, we are told that [tex]\( y \)[/tex] is directly proportional to [tex]\( x \)[/tex]. Mathematically, this relationship can be expressed as:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
### Step 1: Determine the constant of proportionality [tex]\( k \)[/tex]
We are given that [tex]\( y = 14 \)[/tex] when [tex]\( x = 2 \)[/tex]. Substituting these values into the proportionality equation:
[tex]\[ 14 = k \cdot 2 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{14}{2} \][/tex]
[tex]\[ k = 7 \][/tex]
### Step 2: Calculate the value of [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex]
Now that we have the proportionality constant [tex]\( k = 7 \)[/tex], we can find [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex]. Using the equation [tex]\( y = kx \)[/tex]:
[tex]\[ y = 7 \cdot 3 \][/tex]
[tex]\[ y = 21 \][/tex]
### Step 3: Calculate the value of [tex]\( x \)[/tex] when [tex]\( y = 20 \)[/tex]
We need to find [tex]\( x \)[/tex] when [tex]\( y = 20 \)[/tex]. Using the same relationship [tex]\( y = kx \)[/tex] and substituting [tex]\( y = 20 \)[/tex] and [tex]\( k = 7 \)[/tex]:
[tex]\[ 20 = 7x \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{20}{7} \][/tex]
[tex]\[ x \approx 2.857 \][/tex]
### Summary of Results
a) The value of [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex] is [tex]\( 21 \)[/tex].
b) The value of [tex]\( x \)[/tex] when [tex]\( y = 20 \)[/tex] is approximately [tex]\( 2.857 \)[/tex].
These results confirm our understanding and application of direct proportionality.
