Eliminating nonlinear expressions Elimination can remove repeated nonlinear terms as well as linear terms. A factor may be divided out only after its nonzero status is established. Adding x2+y2=A and x2-y2=B removes y2. Each square can allow either sign; only the original equations decide whether signs are linked. Look for equal terms to add or subtract. Factor when useful. If division would remove a possibly zero factor, solve the zero case first, then the nonzero case. Check complete pairs. Example For xy=0 and x2+y2=4: either x=0, giving y=+-2, or y=0, giving x=+-2. There are four pairs. Check: dividing xy=0 by x silently loses (0,2) and (0,-2). Signs are independent for two separate square equations, but xy>0 links their signs.

Elimination can remove repeated nonlinear terms as well as linear terms. A factor may be divided out only after its nonzero status is established.

Adding \(x^2+y^2=A\) and \(x^2-y^2=B\) removes \(y^2\). Each square can allow either sign; only the original equations decide whether signs are linked.

Look for equal terms to add or subtract. Factor when useful. If division would remove a possibly zero factor, solve the zero case first, then the nonzero case. Check complete pairs.

For \(xy=0\) and \(x^2+y^2=4\): either \(x=0\), giving \(y=\pm2\), or \(y=0\), giving \(x=\pm2\). There are four pairs.

Check: dividing \(xy=0\) by \(x\) silently loses \((0,2)\) and \((0,-2)\). Signs are independent for two separate square equations, but \(xy>0\) links their signs.