by Guest » Wed May 15, 2019 11:32 am
[tex]]\frac{P_1V_1}{T_1}= \frac{P_2V_2}{T_2}[/tex]
To solve for any one unknown "undo" whatever is done to it,
[tex]P_2[/tex] has been multiplied by [tex]V_2[/tex] and divided by [tex]T_2[/tex]. To "undo" those divide both sides by [tex]V_2[/tex] and multiply by [tex]T_2[/tex]. And, of course, do that to both sides:
[tex]P_2= \left(\frac{P_1V_1}{T_1}\right)\frac{T_2}{V_2}= \frac{P_1V_1T_2}{T_1V_2}[/tex] as you have.
Solving for [tex]T_2[/tex] is just a little more complicated because [tex]T_2[/tex] is in the denominator. To undo that invert the fraction: [tex]\frac{1}{\frac{1}{x}}= x[/tex]. Inverting both sides [tex]\frac{T_2}{P_2V_2}= \frac{T_1}{P_1V_1}[/tex]. Now multiply both sides by [tex]P_2V_2[/tex]: [tex]T_2= \frac{T_1P_2V_2}{P_1V_1}[/tex].