Given that [tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex], we can start by representing the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] with an equation.
### Part (a): Establishing the Equation
1. Understanding Inverse Proportionality:
- When [tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex], it means that [tex]\( y = \frac{k}{x} \)[/tex], where [tex]\( k \)[/tex] is a constant.
2. Finding the Constant [tex]\( k \)[/tex]:
- We are given that [tex]\( y = 7 \)[/tex] when [tex]\( x = 9 \)[/tex].
- Substitute these values into the inverse relationship equation to find the constant [tex]\( k \)[/tex]:
[tex]\[
7 = \frac{k}{9}
\][/tex]
- To find [tex]\( k \)[/tex], multiply both sides of the equation by 9:
[tex]\[
k = 7 \times 9
\][/tex]
[tex]\[
k = 63
\][/tex]
3. Formulating the Equation:
- Now that we have [tex]\( k \)[/tex], the equation connecting [tex]\( y \)[/tex] and [tex]\( x \)[/tex] can be written as:
[tex]\[
y = \frac{63}{x}
\][/tex]
### Part (b): Finding [tex]\( y \)[/tex] When [tex]\( x = 21 \)[/tex]
1. Substituting [tex]\( x = 21 \)[/tex] into the Equation:
- Use the equation [tex]\( y = \frac{63}{x} \)[/tex] to find [tex]\( y \)[/tex] when [tex]\( x = 21 \)[/tex]:
[tex]\[
y = \frac{63}{21}
\][/tex]
2. Perform the Division:
- Calculate the value:
[tex]\[
y = \frac{63}{21} = 3
\][/tex]
### Summary:
- Equation Connecting [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
y = \frac{63}{x}
\][/tex]
- Value of [tex]\( y \)[/tex] When [tex]\( x = 21 \)[/tex]:
[tex]\[
y = 3
\][/tex]
