If x − y = 7 and xy = 9,

We have:

$(x-y)^{2}=x^{2}-2 x y+y^{2}$

$\Rightarrow x^{2}+y^{2}=(x-y)^{2}+2 x y$

$\Rightarrow x^{2}+y^{2}=7^{2}+2 \times 9 \quad(\because x-y=7$ and $x y=9)$

$\Rightarrow x^{2}+y^{2}=7^{2}+2 \times 9$

$\Rightarrow x^{2}+y^{2}=49+18$

$\Rightarrow x^{2}+y^{2}=67$