q=-20p +100

is the demand curve

graphically the same as

y =-x/20 + 5

q = 50+10p

p = q/10 -5

with a $1 tax shift that line up by 1

p = q/10-4

same as

y = x/10 -4

-x/20 +5 = x/10 -4

-x +100 = 2x -80

3x = 180

x = 60

y = 60/10-4 = 6-4 = $2

new equilibrium price = $2

new equilibrium quantity sold & purchased = 60

old pretax equilibrium was when

-x/20 +5 = x/10 -5

-x +100 = 2x -100

3x =200

x = 200/3 = 66 2/3 = quantity sold & purchased

y = x/10-5 = 200/3/10 -5 =20/3 -5 = 6 2/3-5 = 1 2/3 = $1.67

the price rose 33 cents from $1.67 to $2

and quantity demand & supplied decreased from 66 2/3 to 60

33 cents of the $1 tax was passed on to the consumer

the suppliers only paid 67 cents of the $1 tax

incidence of the tax on the consumer was 33 cents rounded off to nearest cent.

set the two equations equal and solve for x and y

The incidence on consumers is also reflected in the change in consumer surplus, the area of a triangle with height 5-2 =3 and base = 60. A=bh/2 = 60(3)/2 = $90 = gain in trade to the consumer at the new equilibrium price

at the old equilibrium price the area of that triangle, above the price line below the demand curve to the left of the y axis = (200/3)(20/3)/2 = (100/3)(5-5/3) = 1000/9 = $111.11

consumers lost 111.11-90 = $21.11 loss in consumer surplus

with the $1 tax on suppliers, price went up and purchases went down, leaving the net gain in trade as a loss to consumers of $21.11

BUT if the tax is a "lump sum tax" then the equilibrium price and quantity remain the same, with no part of the tax passed on to the consumer and no change in consumer surplus. All the tax is paid by the supplier. It come out of profits only.