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# Help solving this three-part word problem?

At \$0.49 per bushel, the daily supply for wheat is 305 bushels, and the daily demand it 515 bushels.  When the price is raised to \$0.73 per bushel, the daily supply increases to 545 bushels, and the daily demand decreases to 435 bushels.  Assume that the price-supply and price-demand equations are linear.

A.) Find the price-supply equation
B.) Find the price-demand equation
C.) Find the equilibrium price and equilibrium quantity

I'm particularly having trouble with part C. but if you could work this entire problem out for me I would appreciate it!

### 2 Answers by Expert Tutors

William S. | Experienced scientist, mathematician and instructor - WilliamExperienced scientist, mathematician and...
4.4 4.4 (10 lesson ratings) (10)
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A.)  Since the price supply equation is linear, it will have the general form

y = ax + b

Let y = the number of bushels and x the price per bushel:

y = a(\$0.73) + b ⇒ 545 bushels = (\$0.73)a + b
y = a(\$0.49) + b ⇒ 305 bushels = (\$0.49)a + b

Subtract the two equations

(545 bushels - 305 bushels) = a(\$0.73 - \$0.49)

240 bushels = a(\$0.24)

∴ a = (240 bushels)/(\$.24) = 1000 bushels per \$

To find the value of b, plug the value of a into either of the original equations:

y = 545 = (1000)(\$0.73) + b

b = 545 - 730 = -185

So the price-supply equation may be written as: y = (1000)x - 185

B.)   For the price-demand equation use the same procedure

y = a(\$0.73) + b ⇒ 435 bushels = (\$0.73)a + b
y = a(\$0.49) + b ⇒ 515 bushels = (\$0.49)a + b

(435 bushels - 515 bushels) = a(\$0.73 - \$0.49)

-80 bushels = (\$0.24)a

a = (-80 bushels)/(\$0.24) = -333.33

435 bushels = (\$0.73)(-333.33) + b

b = 435 bushels - (\$0.73)(-333.33) = 678.33

The price-demand equation is: y = -333.33x + 683.33

C.)   The equilibrium price and quantity are at the point where these two lines intersect.

y = (1000)x - 185
y = -333.33x + 683.33

1000x - 185 = -333.333x + 683.333

x(1000 + 333.333) = 683.333 + 185

1333.333*x = 868.333

x = (683.333)/(1333.333) = \$0.65 (equilibrium)

y = (1000)(\$0.65) - 185 = 466.25 bushels (equilibrium)

Courtnee,

I should have mentioned yesterday that we could have just as easily allowed the number of bushels to be the independent variable x and the price per bushel be the dependent variable y.  That's one of the nice things about linear equations.
Steve S. | Tutoring in Precalculus, Trig, and Differential CalculusTutoring in Precalculus, Trig, and Diffe...
5.0 5.0 (3 lesson ratings) (3)
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A.) Points for price-supply, (p,s): (\$0.49/b, 305 b), (\$0.73/b, 545 b)

s(p) - 305 = ((545-305)/(0.73-0.49))(p - 0.49)
s(p) - 305 = (240/(0.24))(p - 0.49)
s(p)  = 1000(p - 0.49) + 305
s(p) = 1000p - 490 + 305
s(p) = 1000p - 185 b

B.) Points for price-demand, (p,d): (\$0.49/b, 515 b), (\$0.73/b, 435 b)

d(p) - 515 = ((515-435)/(0.49-0.73))(p - 0.49)
d(p) - 515 = (80/(-0.24))(p - 0.49)
d(p) = -(1000/3)(p - 0.49) + 515
d(p) = -1000p/3 + 490/3 + 1545/3
d(p) = -1000p/3 + 2035/3 b

C.) Use Substitution:
1000p - 185 = -1000p/3 + 2035/3
Multiply by 3:
3000p - 555 = -1000p + 2035
4000p = 555 + 2035 = 2590
p = 259/400 = 2.59/4 = \$0.6475/b

s(\$0.6475/b) = d(\$0.6475/b) = 1000(0.6475) - 185 = 647.5 -185 = 462.5 b