Can we solve these simultaneous equations that involve reciprocals?

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What if the question had asked for the solution to the pair of equations \[\begin{gather*} ax+by=1\\ \frac{a}{x}+ \frac{b}{y}=1 \end{gather*}\]

instead?

You could use the GeoGebra applet below to explore this. Here are some questions you might consider.

1. What curve does each equation generate?
2. How many solutions are there to this pair of simultaneous equations for different values of \(a\) and \(b\)?
   
3. Are the solutions always in the same quadrant?
4. If there are two solutions, can they lie on opposite parts of the curve?