Let’s use the letters w, x, y, z to represent numbers:
wBa(NO₃)₂+xNa₃PO₄➝yBa₃(PO₄)₂+zNaNO₃
Now we can equate the letters:
w=3y (for Ba), 2w=z (for nitrate group NO₃), 3x=z (for Na) and x=2y (for phosphate group PO₄).
So, z=3x=2w=6y. The largest coefficient is 6, belonging to y, so let y=1, then z=6, x=2 and w=3.
That gives us:
3Ba(NO₃)₂+2Na₃PO₄➝Ba₃(PO₄)₂+6NaNO₃
This balances. I’m not a chemist so I don’t know if that’s how these chemicals would react but I guess the acid groups contain strong bonds that won’t be broken in the reaction.