High School

[tex] y [/tex] is inversely proportional to [tex] x [/tex].

When [tex] y = 6 [/tex], [tex] x = 8 [/tex].

a) Work out an equation connecting [tex] y [/tex] and [tex] x [/tex].

b) Work out the value of [tex] y [/tex] when [tex] x = 12 [/tex].

Answer :

Sure, let's solve the problem step-by-step!

When we say that [tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex], it means that as one increases, the other decreases proportionally. Mathematically, this relationship can be expressed as:
[tex]\[ y \times x = k \][/tex]
where [tex]\( k \)[/tex] is a constant.

### a) Finding the equation connecting [tex]\( y \)[/tex] and [tex]\( x \)[/tex].

We are given that when [tex]\( y = 6 \)[/tex], [tex]\( x = 8 \)[/tex]. To find the constant [tex]\( k \)[/tex], we substitute these values into the equation:
[tex]\[ 6 \times 8 = k \][/tex]
[tex]\[ k = 48 \][/tex]

Now that we have [tex]\( k \)[/tex], the equation that connects [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is:
[tex]\[ y = \frac{48}{x} \][/tex]

### b) Finding the value of [tex]\( y \)[/tex] when [tex]\( x = 12 \)[/tex].

Using the equation we found, substitute [tex]\( x = 12 \)[/tex] into the equation:
[tex]\[ y = \frac{48}{12} \][/tex]
[tex]\[ y = 4 \][/tex]

So, when [tex]\( x = 12 \)[/tex], the value of [tex]\( y \)[/tex] is 4.