Relative Humidity Problem

I have been assigned to solve the problem: the ongoing temperature of the room $T_1 = 7$; RH is 85% and I am asked to figure out how much the temperature must be increased in order that RH would drop to 70% or less.

I would like to know what I did wrong – why I could not find this second temperature. I can infinitely use the definition of RH and Antoine equation, but never find $T_2$ in the way. Why? Because I do not see any connection between parameters $p_1, p_{sat_1}, v_1, v_{sat_1}, p_2, p_{sat_2}, v_2, v_{sat_2}$. Knowing $T_1$ I can find $p_1$ and then $v_1$; and also the temperature at which the air shall be saturated with water vapour: $T_{d_1}$. Knowing them does not solve the risen problem. That means that I do not understand something about how these parameters are connected between themselves.

The only possibility for me is to look into what happens in the room upon the rising the temperature up to $T_2$. Will the RH drop? Not necessarily, because the more environment’s temperature, the more moisture the air can store, so boosting RH. But okay we did somehow rise $T$ up to $T_2$ and $varphi_2 = 0.7$. Definitely, $T_{d_2} ne T_{d_1}$; maybe $p_{sat_2} = p_{sat_1}$? I think no as well, because both parameters depend on the specific temperature.

What do I misunderstand? Maybe a quantity gets preserved? And by it I should solve the problem. I am confused. Volumes are not given. Only the concurrent temperature of the room and RH; then they boost temperature and tell that RH is 0.7.

NOTATION

$T$ is temperature,

$varphi$ is RH

$v$ is the water vapor contained in the air

$p$ is the partial water vapour pressure

$p_{sat}$ when the air gets saturated with the water vapour

$v_{sat}$ is the amount of the water vapour contained in the saturated air.