Question 55054
You have one equation that defines demand
{{{p^2 + q^2 = 169}}}

and one that defines supply
{{{p = q+7}}}

You are right that equilibrium happens when supply equals demand. 
However, what you want to do is substitute the second equation into the first one.

If we do that, we get
{{{(q+7)^2 + q^2 = 169}}}
I'll FOIL that first term
{{{q^2+14q+49+q^2 = 169}}}
Combine like terms
{{{2q^2 + 14q + 49 = 169}}}
Moving the 169 to the left
{{{2q^2 +14q -120 = 0}}}
Divide both sides by 2
{{{q^2 + 7q - 60 = 0}}}
I can solve this quadratic equation by any number of ways. I'll use factoring
Factoring it out, I get
{{{(q+12)(q-5)=0}}}
Solving, I get
{{{q  = -12}}}
and
{{{q = 5}}}

We can toss -12 out because a negative quantity is nonsense for this problem. 

I can then put {{{q = 5}}} into our supply equation

{{{p = q + 7 }}}
{{{p = 5+7}}}
{{{p = 12}}}

Ok, equilibrium price is 7 at a quantity of 5.