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 $(wr+x)\div(yr+z)=1$

Question

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.

Answer 1

One way to prove uniqueness, and is a way that can be used here, is to find a closed expression for the value you are searching.

In this case by isolating r (and being careful with divisions by 0, i.e. justifying that what you are dividing is not 0) you will get a closed expression for r. Meaning that the solution is unique (it can only be that exact value).

Hope I was useful.

Answer 2

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 $r=s$.