[Compare your answers to (a) and (b) with the answers for (a) and (b) in question 2.
Also, retry (b) and (c) with the 20 in (i) changed to 30, and compare your answers
with that for 3.(d).]
4.
Solving, you should have
x
= (
a
1
–
a
2
)/(
b
2
–
b
1
), or equivalently,
x
= (
a
2
–
a
1
)/(
b
1
–
b
2
),
with y = (
a
1
b
2
–
a
2
b
1
)/(
b
2
–
b
1
) = (
a
2
b
1
–
a
1
b
2
)/(
b
1
–
b
2
).
[These solution values for
x
and
y
apply generally. As an exercise, you may therefore
want to try substituting in the values from some of the cases in questions 13, to con
M
ATH
M
ODULE
S
olutions to
Exercises
3
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firm that you get the correct answers. Do
not
feel obliged to memorize these formu
las; it’s almost always easier to work out the questions “from scratch.” What the for
mulas show, however, is that the solutions to every 2equation linear system have
exactly the same form.
The formulas also show why
inconsistent
equations have no solution: the denom
inator for both
x
and
y
solutions is (
b
2
–
b
1
), or (
b
1
–
b
2
), where
b
1
and
b
2
are the slopes
of the 2 equations in the system. When the 2 lines are parallel,
b
1
=
b
2
, and so the
denominator is zero and the solution values for
x
and
y
are thus nonexistent.
5. (a)
P =
$20/unit,
Q =
4000 units. (b)
P =
$20/unit,
Q =
400 units. (c) Rearranging, (i)
P =
20 – 0.5
Q
D
and (ii)
P =
30 +
Q
S
. [This equation has a
formal
or
mathematical
solu
tion, with
P =
$23.33/unit and
Q =
– 6 2/3 units. If we are dealing with
standard
com
modities, however, then the meaning of
negative
quantities is unclear. Often, even if
we do not state it explicitly, we assume that
P ≥
$0 and that
Q ≥
$0. In the present
case, since there is zero quantity supplied at a price of $30/unit and no one is will
ing to pay more than $20/unit for positive amounts of the good, we can say that the
nonnegativity constraint (
Q ≥
$0) is binding, and that the quantity exchanged is
zero.
(This question was a “trick” question: its purpose is to remind us that even when our
equations “work out”
mathematically
, we still need to ensure that they make
econom
ic
sense.)]
6. (a) Given
P
S
= 20 + 0.1
Q
S
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 Fall '12
 Danvo
 Harshad number, Elementary algebra, Sal, Math Module

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