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Math 116.01 - Quiz 4
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No notes. Please show enough work to demonstrate your reasoning.
(1) Sove the linear system
x
+ 4y + 3z = 2
−x − y + z = 1
−2x − 8y − 6z = −1
First, we’ll solve the first equation for x:
x = 2 − 4y − 3z.
Next, we’ll eliminate x from the other two equations to get a smaller system:
−(2 − y4 − 3z) − y + z = 1 ⇒
−2(2 − 4y − 3z) − 8y − 6z = −1 ⇒
3y + 4z = 3
−4 = 1.
Because there is no choice of y and z that will make the equation −4 = 1 true, there is no
solution to this smaller system. And since there is no solution to the smaller system, there
is no solution to the original system, either.
(MORE ON BACK)
1
2
(2) Suppose supply and demand are determined by the following equations:
1 2
supply: p = 10
q − 2q + 20
demand: p = 15 − 41 q
(a) Find the price and quantities of any market equilibria.
Since both equations in this system are already solved for p, we can eliminate p by
setting them equal:
15 − 14 q =
0=
1 2
10 q
1 2
10 q
2
− 2q + 20
−
7
4
+5
0 = 2q − 35q + 100
p
35 ± (35)2 − 4(2)(100)
q=
2(2)
≈ 13.90388, 3.5961.
When q = 13.90388, p is given by p = 15 − 14 q = 11.524. When q = 3.5961, p is given
by p = 15 − 14 q = 14.1009. So there are actually two equilibria.
(b) Suppose the government imposes a tax of $1.50 per unit. Modify the system above to
reflect this fact. (You should write both the demand and supply equations for the new
system. Do not solve the new system)
Recall that when we impost a tax, the suppliers don’t get the full sale price p. They
only get the sale price minus the tax, p − 1.5. So in the supply equation, we must
replace p with p − 1.5. The buyers don’t care what makes up that sale price (tax,
manufacturing cost, producer profits, etc.), so the demand equation does not change.
This gives the system
1 2
q − 2q + 20
supply: p − 1.5 = 10
demand:
p = 15 − 14 q