# For all integers $w, x, y, z$ with $w\neq{y}$ and $wz-xy\neq0$, prove that there exists a unique rational number $r$ such that $(wr+x)\div(yr+z)=1$

For all integers $$w, x, y, z$$ with $$w\neq{y}$$ and $$wz-xy\neq0$$, prove that there exists a unique rational number $$r$$ such that $$(wr+x)\div(yr+z)=1$$

How do I prove uniqueness? I know to show that there exists a number all I need to do is use an example.

• Assume $(wr + x)/(yr+z)=(ws+x)/(ys+z)$ and then show $r=s$ under the assumptions. – J. W. Tanner Jan 29 at 19:06

To prove uniqueness, assume $$(wr + x)/(yr+z)=(ws+x)/(ys+z).$$ Cross-multiplying, $$(wr+x)(ys+z)=(ws+x)(yr+z),$$ so $$wrys+wrz+xys+xz=wsyr+wsz+xyr+xz,$$ so $$wrz+xys=wsz+xyr,$$ so $$wrz-wsz=xyr-xys,$$ i.e., $$(wz-xy)r=(wz-xy)s.$$ Under the assumption $$wz-xy \ne 0$$, this means $$r=s$$.