$x\equiv 11\phantom{\rule{1em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}84)$

$x\equiv 23\phantom{\rule{1em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}36)$

I have the bulk of the work done for this;

$x=11+84j$

$x=23+36k$

$\Rightarrow 11+84j\equiv 23\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}36)$

$\Rightarrow 84j\equiv 12\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}36)$

$\Rightarrow 12j\equiv 12\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}36)$

$\Rightarrow j\equiv 1\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}36)$

$\Rightarrow j=1+36n$

Thus this system is true for any $x$ of the form

$x=11+84(1+36n)=95+3024n$

$x\equiv 23\phantom{\rule{1em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}36)$

I have the bulk of the work done for this;

$x=11+84j$

$x=23+36k$

$\Rightarrow 11+84j\equiv 23\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}36)$

$\Rightarrow 84j\equiv 12\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}36)$

$\Rightarrow 12j\equiv 12\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}36)$

$\Rightarrow j\equiv 1\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}36)$

$\Rightarrow j=1+36n$

Thus this system is true for any $x$ of the form

$x=11+84(1+36n)=95+3024n$