To solve this problem, we need to understand the concept of inverse variation. When we say that [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex], it means that [tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex]. This can be mathematically represented by the equation:
[tex]\[ y = \frac{k}{x} \][/tex]
Here, [tex]\( k \)[/tex] is the variation constant that we need to determine.
We are given the following information:
- [tex]\( y = 6 \)[/tex]
- [tex]\( x = 7 \)[/tex]
To find the variation constant [tex]\( k \)[/tex], we can substitute the given values of [tex]\( y \)[/tex] and [tex]\( x \)[/tex] into the inverse variation equation:
[tex]\[
6 = \frac{k}{7}
\][/tex]
To solve for [tex]\( k \)[/tex], we multiply both sides of the equation by 7:
[tex]\[
6 \times 7 = k
\][/tex]
[tex]\[
k = 42
\][/tex]
So, the variation constant [tex]\( k \)[/tex] is 42.
Next, we need to write the equation that represents this inverse variation. Using the value of [tex]\( k \)[/tex] we just found and substituting it back into the inverse variation equation:
[tex]\[
y = \frac{42}{x}
\][/tex]
Therefore, the equation of variation is:
[tex]\[
y = \frac{42}{x}
\][/tex]
In summary:
- The variation constant is [tex]\( k = 42 \)[/tex].
- The equation of variation is [tex]\( y = \frac{42}{x} \)[/tex].
