NTC thermistor series/parallel resistor calculation for MCP73833 linear Li-Ion charger
You're a good part of the way there. Simply subtract the two equations and you get

\$R_{T1}^2 -29R_{T1}+m = 0\$ 

where  \$ m = 24\cdot5 - \frac{19}{1/R_{HOT}-1/{R_{COLD}}}\$ (units of k\$\Omega^2\$)

So \$R_{T1}= \frac{29 \pm \sqrt{29^2 -4m}}{2}\$ (standard solutions to a quadratic) 

Pick the correct answer for RT1 (minus) and you can then subsitute RT1 back into either of the first two equations to get (say) 

\$R_{T2} = \frac{1}{\frac{1}{5K-R_{T1}}- \frac{1}{R_{HOT}}}\$

(If you end with a negative answer for either resistor, you picked the wrong answer- assuming there is a solution for that thermistor).