e/d
= 30/ 5 = 6, the slope is
–
(
c/d
) = –3/5, and the horizontal intercept is
e/c
= 30/3 = 10. The
standard form of the equation would be
y = e/d –
(
c/d
)
x
= 6 – (3/5)
x
.
The equation 3
x
+ 5
y
= 30 could be a budget constraint, for someone spending $30,
facing prices of $3/unit for
x
and $5/unit for
y
, with
x
and
y
representing the number
of units of the two goods purchased. In this case, the vertical intercept is the maximum
number of units of
y
that can be purchased (= 30/5 = 6), the horizontal intercept is the
maximum number of units of
x
that can be purchased (= 30/3 = 10), and the slope (=
–3/5) indicates that at any point, if 3/5 of a unit of
y
is
given up
, 1
additional
unit of
x
can
be purchased with the money saved. Regardless of the
economic
interpretation we give
to a particular linear function, however, the above
mathematical
relationships will hold
true.
2. Exercises
1. Convert each of the following equations into the form
y
=
a
+
b
x, where
a
is the ver
tical intercept and
b
is the slope, give the horizontal intercept for each, and graph
all
of them on a single diagram.
(a) 3 = (
y
– 6)/(
x
– 6)
(b) 3 = (
y
+ 6)/(6 –
x
)
(c) 9
x
+ 3
y
= 36
(d) 9
x
– 3
y
= 36
(e)
x
= 4 + (1/3)
y
(f)
x
= 4 – (1/3)
y
(g)
x
= –4 + (1/3)
y
(h)
x
= –4 – (1/3)
y
2. Graph the following equations on the same diagram. ±or each equation, give the
value of
y
when
x =
10 and
x
= 20
and
the value of
x
when
y
= 5 and
y
= 10.
(a)
y
= 10 – 0.5
x
(b)
y
= 10 –
x
(c)
y
= 10 – 2
x
(d)
y
= 20 –
x
(e)
y
= 20 – 0.5
x
M24
MATH MODULE 2: LINEAR EQUATIONS
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View Full Document3. Give the equations for the following in slopeintercept form,
y
=
a
+
b
x, graph them
all on a single diagram and give the horizontal and vertical intercepts where appro
priate:
(a) Slope = +3, line passes through (10, 40)
(b) Slope = +2, line passes through (5, 30)
(c) Slope = +1, line passes through (–30, 0)
(d) Slope = 0, line passes through (20, 40)
(e) Slope = –1, line passes through (40, 10)
(f) Slope = inFnity [vertical], line passes through (10, 10).
4. Given the following data for sets of 2 points on some straight lines, calculate the
slopes of the lines and give their equations in slopeintercept form,
y
=
a
+
b
x, and
graph them in a single diagram:
(a)
A
1
(20, 10),
A
2
(0, 30)
(b)
B
1
(10, 12.5),
B
2
(40, 5)
(c)
C
1
(10, 5),
C
2
(30, 15)
(d)
D
1
(10, 0),
D
2
(15, 5)
(e)
E
1
(5, 10),
E
2
(15, 10)
MATH MODULE 2: LINEAR EQUATIONS
M25
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 Fall '12
 Danvo
 Slope, Supply And Demand, Euclidean geometry

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