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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.
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.
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