There are answers which switch the demand equation, but I$\text{'}$m $95\%$ certain the supply equation needs to change based on my classes.

The issue though, when I change the supply equation to $20+20(1\mathrm{.}12$P$),$ I get Qs$1=20+22\mathrm{.}4$P and when I try to find the equilibrium price at Qs$1=$Qd which leads to $20+22\mathrm{.}4$P$=140-3$P$,$ P$=4\mathrm{.}7244,$ a lower equilibrium price than the original $(5\mathrm{.}217).$ This also leads to a new equilibrium quantity at $125\mathrm{.}8266$ which is higher than the original equilibrium quantity at $124\mathrm{.}35.$

However, from my knowledge, If a tax is imposed, the new equilibrium price $($$$4\mathrm{.}7244)$ should increase from the original $($$$5\mathrm{.}217)$ and the equilibrium quantity $(124\mathrm{.}34)$ should decrease not increase $(125\mathrm{.}8266).$ I believe I have made an error in imposing the $12\%$ tax into the equation; therefore, my answers are coming out incorrect.

If you could first solve this properly changing the supply equation, then add another step by step on what would happen if you imposed the tax on the demand equation, I would be grateful as that would help me see the difference in the equations and effects to placing the tax on different equations for question b$.$ The main solution I need would be changing the supply equation, but it'd be really helpful if you could include the new demand equation and equilibrium price$/$quantity if it affected demand.

I need someone to solve this correctly and a very indepth step by step guide as I$\text{'}$m struggling to grasp the concept and solve this based on a percentage tax rather than a dollar amount. If you could please include graph illustrations for question C that'd be greatly appreciated, thank you for your help!!