Choice of 2 formulae. The first is a fraction of the form y=(r+x)(1-rx). The second is of the form y=sqrt[(1-x)/(1+x) ]. Rearrange to make x the subject.
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", "advice": "$\\begin{align}y&=\\frac{r+x}{1-rx}\\\\ y(1-rx) &= r+x\\qquad\\text{multiply both sides by }(1-rx)\\\\y-ryx &= r+x\\qquad\\textrm{expand LHS}\\\\y-r &=x+ryx\\qquad\\text{rearrange}\\\\y-r&=x(1+ry)\\qquad\\text{factorise RHS}\\\\\\frac{y-r}{1+ry}&=x\\qquad\\text{divide by }(1+ry)\\\\x&=\\frac{y-r}{1+ry} \\end{align}$
\n$\\begin{align} y&=\\sqrt{\\frac{x-1}{x+1}}\\\\y^2&=\\frac{x-1}{x+1}\\qquad\\text{square}\\\\y^2(x+1)&=x-1\\qquad\\text{multiply by }(x+1)\\\\y^2x+y^2&=x-1\\qquad\\text{expand LHS}\\\\y^2x-x&=-y^2-1\\qquad\\text{rearrange}\\\\x(y^2-1)&=-y^2-1\\qquad\\text{factorise LHS}\\\\x&=\\frac{-y^2-1}{y^2-1} \\end{align}$
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