Let's solve the problem step-by-step.
### Part (a) - Work out an equation connecting [tex]\( y \)[/tex] and [tex]\( x \)[/tex]
Since [tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex], we can express their relationship using the equation:
[tex]\[ y = \frac{k}{x} \][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
We are given the values [tex]\( y = 7 \)[/tex] and [tex]\( x = 9 \)[/tex]. We can use these values to find the constant [tex]\( k \)[/tex].
Plugging the given values into the equation:
[tex]\[ 7 = \frac{k}{9} \][/tex]
To find [tex]\( k \)[/tex], we multiply both sides of the equation by [tex]\( 9 \)[/tex]:
[tex]\[ k = 7 \times 9 = 63 \][/tex]
So, the equation connecting [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is:
[tex]\[ y = \frac{63}{x} \][/tex]
### Part (b) - Work out the value of [tex]\( y \)[/tex] when [tex]\( x = 21 \)[/tex]
Now we need to find the value of [tex]\( y \)[/tex] for [tex]\( x = 21 \)[/tex] using the equation we derived in part (a):
[tex]\[ y = \frac{63}{x} \][/tex]
Substituting [tex]\( x = 21 \)[/tex]:
[tex]\[ y = \frac{63}{21} \][/tex]
Simplifying the right-hand side:
[tex]\[ y = 3 \][/tex]
So, the value of [tex]\( y \)[/tex] when [tex]\( x = 21 \)[/tex] is:
[tex]\[ y = 3 \][/tex]
### Summary
1. The equation connecting [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is [tex]\( y = \frac{63}{x} \)[/tex].
2. When [tex]\( x = 21 \)[/tex], [tex]\( y \)[/tex] is [tex]\( 3 \)[/tex].
