jmc

In general, if we have fractions $\frac{a}{b}=\frac{c}{d},$ then $$\frac{a}{b}=\frac{c}{d}=\frac{a+c}{b+d}.$$To see why, let $k=\frac{a}{b}=\frac{c}{d}.$ Then $a=kb$ and $c=kd,$ so $$\frac{a+c}{b+d}=\frac{kb+kd}{b+d}=k.$$Applying this here, we get $$\frac{7}{x+y}=\frac{11}{z-y}=\frac{7+11}{(x+y)+(z-y)}=\frac{18}{x+z}.$$Hence, $k=\boxed{18}.$