Solution:
Given that z varies directly as x and inversely as y, this is expressed mathematically as
[tex]z\propto x\propto\frac{1}{y}[/tex]
Introducing a proportionality constant k, we have
[tex]\begin{gathered} z=k\frac{x}{y} \\ \Rightarrow k=\frac{zy}{x}\text{ ---- equation 1} \end{gathered}[/tex]
When z = 123, x=6, and y=6 , k is evaluated to be
[tex]\begin{gathered} k=\frac{zy}{x}\text{ } \\ =\frac{123\times6}{6} \\ \Rightarrow k=123 \end{gathered}[/tex]
Thus, equation 1 becomes
[tex]123=\frac{zy}{x}[/tex]
To evaluate z when x =7 and y=8, we substitute the respective values of x and y into the above equation.
Thus,
[tex]\begin{gathered} 123=\frac{zy}{x} \\ \Rightarrow123=\frac{z\times8}{7} \\ \text{make z the subject of the formula} \\ z\times8=123\times7 \\ \text{divide both sides by 8} \\ \frac{z\times8}{8}=\frac{123\times7}{8} \\ z=\frac{123\times7}{8} \\ =\: 107.625 \\ \Rightarrow z\approx107.63\text{ (nearest hundredth)} \end{gathered}[/tex]
Hence, the value of z when x =7 and y =8 is
[tex]107.63\text{ (nearest hundredth)}[/tex]
