## Joint Variation

 1.3 Joint Variation

 Definition joint variation In general, for a combined variation, $$y$$ varies directly as $$x^m$$ and inversely as $$z^n$$ can be written as \begin{aligned}x\propto\frac{x^m}{z^n}\end{aligned}\hspace{1mm}\text{(variation relation)} or \begin{aligned} x=\frac{kx^m}{z^n} \end{aligned} \hspace{1mm} \text{(equation relation)} such that \begin{aligned} m&=1,2,3,\frac{1}{2},\frac{1}{3},\hspace{1mm} \\\\n&=1,2,3,\frac{1}{2},\frac{1}{3} \end{aligned} and $$k$$ is a constant.
 Example 3 Given that $$y$$ varies directly as then square of $$x$$  varies inversely as square root of $$z$$. If $$y=8$$ when $$x=4$$ and $$z=36$$, express $$y$$ in terms of $$x$$ and $$z$$. Solution: \begin{aligned}\hspace{1mm}& y\propto \frac{x^2}{\sqrt{2}}\implies y = \frac{kx^2}{\sqrt{2}} \dots (1). \end{aligned} Substitute $$y=8$$, $$x=4$$, and $$z=36$$ into $$(1)$$: \begin{aligned}8&=\frac{k4^2}{\sqrt{36}}\implies k=\frac{(8)(6)}{16}\\\\&=3.\\\\ &\therefore y=\frac{3x^2}{\sqrt{z}}. \end{aligned}