Transcript of "Math 1300: Section 4-1 Review: Systems of Linear Equations in Two Variables"
1.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Math 1300 Finite Mathematics
Section 4.1 Review: Systems of Linear Equations in Two
Variables
Jason Aubrey
Department of Mathematics
University of Missouri
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Jason Aubrey Math 1300 Finite Mathematics
2.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Deﬁnition (Systems of Equations in Two Variables)
Given the linear system
ax + by = h
cx + dy = k
A pair of numbers x = x0 , y = y0 [also written as an ordered
pair (x0 , y0 ) is said to be a solution of the system if each
equation is satisﬁed by the pair. The set of all such ordered
pairs is called the solution set for the system. To solve a
system is to ﬁnd its solution set.
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Jason Aubrey Math 1300 Finite Mathematics
3.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Deﬁnition (Systems of Equations in Two Variables)
Given the linear system
ax + by = h
cx + dy = k
A pair of numbers x = x0 , y = y0 [also written as an ordered
pair (x0 , y0 ) is said to be a solution of the system if each
equation is satisﬁed by the pair. The set of all such ordered
pairs is called the solution set for the system. To solve a
system is to ﬁnd its solution set.
We will consider three methods for solving such systems:
graphing, substitution, and elimination by addition. university-logo
Jason Aubrey Math 1300 Finite Mathematics
4.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by graphing:
3x − y = 2
x + 2y = 10
10
8
6
4
2
−4 −2 2 4 6 8 10 university-logo
−2
Jason Aubrey Math 1300 Finite Mathematics
5.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by graphing:
3x − y = 2
x + 2y = 10
10
8
6
4
2
−4 −2 2 4 6 8 10 university-logo
−2
Jason Aubrey Math 1300 Finite Mathematics
6.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by graphing:
3x − y = 2
x + 2y = 10
10
8
6
4
2
−4 −2 2 4 6 8 10 university-logo
−2
Jason Aubrey Math 1300 Finite Mathematics
7.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by graphing:
3x − y = 2
x + 2y = 10
10
8
6
(2, 4)
4
2
−4 −2 2 4 6 8 10 university-logo
−2
Jason Aubrey Math 1300 Finite Mathematics
8.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Deﬁnition (Systems of Linear Equations: Basic Terms)
A system of linear equations is consistent if it has one or more
solutions and inconsistent if no solutions exist. Furthermore, a
consistent system is said to be independent if it has exactly
one solution and dependent if it has more than one solution.
Two systems of equations are equivalent if they have the same
solution set.
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Jason Aubrey Math 1300 Finite Mathematics
9.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Theorem (Possible Solutions to a Linear System)
The linear system
ax + by = h
cx + dy = k
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Jason Aubrey Math 1300 Finite Mathematics
10.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Theorem (Possible Solutions to a Linear System)
The linear system
ax + by = h
cx + dy = k
Must have
Exactly one solution (consistent and independent), or
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Jason Aubrey Math 1300 Finite Mathematics
11.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Theorem (Possible Solutions to a Linear System)
The linear system
ax + by = h
cx + dy = k
Must have
Exactly one solution (consistent and independent), or
No solution (inconsistent), or
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Jason Aubrey Math 1300 Finite Mathematics
12.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Theorem (Possible Solutions to a Linear System)
The linear system
ax + by = h
cx + dy = k
Must have
Exactly one solution (consistent and independent), or
No solution (inconsistent), or
Inﬁnitely many solutions (consistent and dependent).
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Jason Aubrey Math 1300 Finite Mathematics
13.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Theorem (Possible Solutions to a Linear System)
The linear system
ax + by = h
cx + dy = k
Must have
Exactly one solution (consistent and independent), or
No solution (inconsistent), or
Inﬁnitely many solutions (consistent and dependent).
There are no other possibilities.
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Jason Aubrey Math 1300 Finite Mathematics
14.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Graph the equations and ﬁnd the coordinates of any
points where two or more lines intersect. Discuss the nature of
the solution set.
x − 2y = −6
2x + y = 8
x + 2y = −2
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Jason Aubrey Math 1300 Finite Mathematics
15.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
6
4
2
−4 −2 0 2 4
−2
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Jason Aubrey Math 1300 Finite Mathematics
16.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
6
x − 2y = −6
4
2
−4 −2 0 2 4
−2
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Jason Aubrey Math 1300 Finite Mathematics
17.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
6
x − 2y = −6
4 2x + y = 8
2
−4 −2 0 2 4
−2
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Jason Aubrey Math 1300 Finite Mathematics
18.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
6
x − 2y = −6
4 2x + y = 8
x + 2y = −2
2
−4 −2 0 2 4
−2
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Jason Aubrey Math 1300 Finite Mathematics
19.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
6
(2, 4) x − 2y = −6
4 2x + y = 8
x + 2y = −2
2
−4 −2 0 2 4
−2
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Jason Aubrey Math 1300 Finite Mathematics
20.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
6
(2, 4) x − 2y = −6
4 2x + y = 8
(−2, 2)
x + 2y = −2
2
−4 −2 0 2 4
−2
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Jason Aubrey Math 1300 Finite Mathematics
21.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
6
(2, 4) x − 2y = −6
4 2x + y = 8
(−2, 2)
x + 2y = −2
2
No point lies on all
three lines. So, no so-
−4 −2 0 2 4 ( 14 , − 4 )
3 3 lution to this system.
−2
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Jason Aubrey Math 1300 Finite Mathematics
22.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by substitution:
2x + y = 6
x − y = −3
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23.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by substitution:
2x + y = 6
x − y = −3
2x + y = 6
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Jason Aubrey Math 1300 Finite Mathematics
24.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by substitution:
2x + y = 6
x − y = −3
2x + y = 6
y = 6 − 2x now substitute:
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Jason Aubrey Math 1300 Finite Mathematics
25.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by substitution:
2x + y = 6
x − y = −3
2x + y = 6
y = 6 − 2x now substitute:
x − y = −3
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Jason Aubrey Math 1300 Finite Mathematics
26.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by substitution:
2x + y = 6
x − y = −3
2x + y = 6
y = 6 − 2x now substitute:
x − y = −3
x − (6 − 2x) = −3
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Jason Aubrey Math 1300 Finite Mathematics
27.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by substitution:
2x + y = 6
x − y = −3
2x + y = 6
y = 6 − 2x now substitute:
x − y = −3
x − (6 − 2x) = −3
3x − 6 = −3
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Jason Aubrey Math 1300 Finite Mathematics
28.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by substitution:
2x + y = 6
x − y = −3
2x + y = 6
y = 6 − 2x now substitute:
x − y = −3
x − (6 − 2x) = −3
3x − 6 = −3
x =1
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Jason Aubrey Math 1300 Finite Mathematics
29.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system by substitution:
2x + y = 6
x − y = −3
2x + y = 6
y = 6 − 2x now substitute:
x − y = −3
x − (6 − 2x) = −3
3x − 6 = −3
x =1
Substituting we obtain y = 4. So, the solution is the point (1, 4).
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Jason Aubrey Math 1300 Finite Mathematics
30.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Graphing and substitution work well for systems involving
two variables.
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Jason Aubrey Math 1300 Finite Mathematics
31.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Graphing and substitution work well for systems involving
two variables.
However, neither is easily extended to larger systems.
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Jason Aubrey Math 1300 Finite Mathematics
32.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Graphing and substitution work well for systems involving
two variables.
However, neither is easily extended to larger systems.
Elimination by addition is the most important method of
solution.
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Jason Aubrey Math 1300 Finite Mathematics
33.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Graphing and substitution work well for systems involving
two variables.
However, neither is easily extended to larger systems.
Elimination by addition is the most important method of
solution.
It readily generalizes to larger systems and forms the basis
for computer-based solution methods.
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Jason Aubrey Math 1300 Finite Mathematics
34.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Theorem (Operations that Produce Equivalent Systems)
A system of linear equations is transformed into an equivalent
system if
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Jason Aubrey Math 1300 Finite Mathematics
35.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Theorem (Operations that Produce Equivalent Systems)
A system of linear equations is transformed into an equivalent
system if
(A) Two equations are interchanged.
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Jason Aubrey Math 1300 Finite Mathematics
36.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Theorem (Operations that Produce Equivalent Systems)
A system of linear equations is transformed into an equivalent
system if
(A) Two equations are interchanged.
(B) An equation is multiplied by a nonzero constant.
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Jason Aubrey Math 1300 Finite Mathematics
37.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Theorem (Operations that Produce Equivalent Systems)
A system of linear equations is transformed into an equivalent
system if
(A) Two equations are interchanged.
(B) An equation is multiplied by a nonzero constant.
(C) A constant multiple of one equation is added to another
equation.
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Jason Aubrey Math 1300 Finite Mathematics
38.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system using elimination by addition
3u − 2v = 12
7u + 2v = 8
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Jason Aubrey Math 1300 Finite Mathematics
39.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system using elimination by addition
3u − 2v = 12
7u + 2v = 8
3u − 2v = 12
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Jason Aubrey Math 1300 Finite Mathematics
40.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system using elimination by addition
3u − 2v = 12
7u + 2v = 8
3u − 2v = 12
+7u + 2v = 8
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Jason Aubrey Math 1300 Finite Mathematics
41.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system using elimination by addition
3u − 2v = 12
7u + 2v = 8
3u − 2v = 12
+7u + 2v = 8
10u + 0 = 20
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Jason Aubrey Math 1300 Finite Mathematics
42.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system using elimination by addition
3u − 2v = 12
7u + 2v = 8
3u − 2v = 12
+7u + 2v = 8
10u + 0 = 20
u=2
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Jason Aubrey Math 1300 Finite Mathematics
43.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve the linear system using elimination by addition
3u − 2v = 12
7u + 2v = 8
3u − 2v = 12
+7u + 2v = 8
10u + 0 = 20
u=2
Substituting, we obtain v = −3. So the solution is (2,-3)
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Jason Aubrey Math 1300 Finite Mathematics
44.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve using elimination by addition.
3x − 2y = 8
2x + 5y = −1
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Jason Aubrey Math 1300 Finite Mathematics
45.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve using elimination by addition.
3x − 2y = 8
2x + 5y = −1
We multiply the ﬁrst equation by 5 and the bottom equation by 2
and then add.
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Jason Aubrey Math 1300 Finite Mathematics
46.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve using elimination by addition.
3x − 2y = 8
2x + 5y = −1
We multiply the ﬁrst equation by 5 and the bottom equation by 2
and then add.
15x − 10y = 40
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Jason Aubrey Math 1300 Finite Mathematics
47.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve using elimination by addition.
3x − 2y = 8
2x + 5y = −1
We multiply the ﬁrst equation by 5 and the bottom equation by 2
and then add.
15x − 10y = 40
4x + 10y = −2
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Jason Aubrey Math 1300 Finite Mathematics
48.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve using elimination by addition.
3x − 2y = 8
2x + 5y = −1
We multiply the ﬁrst equation by 5 and the bottom equation by 2
and then add.
15x − 10y = 40
4x + 10y = −2
19x = 38
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Jason Aubrey Math 1300 Finite Mathematics
49.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve using elimination by addition.
3x − 2y = 8
2x + 5y = −1
We multiply the ﬁrst equation by 5 and the bottom equation by 2
and then add.
15x − 10y = 40
4x + 10y = −2
19x = 38
1
Now we multiply both sides of this last equation by 19 to obtain
x = 2.
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Jason Aubrey Math 1300 Finite Mathematics
50.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: Solve using elimination by addition.
3x − 2y = 8
2x + 5y = −1
We multiply the ﬁrst equation by 5 and the bottom equation by 2
and then add.
15x − 10y = 40
4x + 10y = −2
19x = 38
1
Now we multiply both sides of this last equation by 19 to obtain
x = 2. Then we substitute back into either of the two original
equations to obtain y = −1. university-logo
Jason Aubrey Math 1300 Finite Mathematics
51.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Example: At $4.80 per bushel, the annual supply for soybeans
in the Midwest is 1.9 billion bushels and the annual demand is
2.0 billion bushels. When the price increases to $5.10 per
bushel, the annual supply increases to 2.1 billion bushels and
the annual demand decreases to 1.8 billion bushels. Assume
that the supply and demand equations are linear.
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Jason Aubrey Math 1300 Finite Mathematics
52.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(a) Find the supply equation.
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Jason Aubrey Math 1300 Finite Mathematics
53.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(a) Find the supply equation.
We wish to ﬁnd an equation of the form p = mq + b where p
represents unit price and q represents quantity demanded.
We have two points (q, p) on the graph of the supply equation:
(1.9, 4.80) and (2.1, 5.10).
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Jason Aubrey Math 1300 Finite Mathematics
54.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(a) Find the supply equation.
We wish to ﬁnd an equation of the form p = mq + b where p
represents unit price and q represents quantity demanded.
We have two points (q, p) on the graph of the supply equation:
(1.9, 4.80) and (2.1, 5.10).
The slope of this line is:
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Jason Aubrey Math 1300 Finite Mathematics
55.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(a) Find the supply equation.
We wish to ﬁnd an equation of the form p = mq + b where p
represents unit price and q represents quantity demanded.
We have two points (q, p) on the graph of the supply equation:
(1.9, 4.80) and (2.1, 5.10).
The slope of this line is:
5.10 − 4.80 0.3 3
m= = =
2.1 − 1.9 0.2 2
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Jason Aubrey Math 1300 Finite Mathematics
56.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Therefore the equation of this line is
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Jason Aubrey Math 1300 Finite Mathematics
57.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Therefore the equation of this line is
3
p − 4.8 = (q − 1.9)
2
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Jason Aubrey Math 1300 Finite Mathematics
58.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Therefore the equation of this line is
3
p − 4.8 = (q − 1.9)
2
3 3
p = q − (1.9) + 4.8
2 2
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Jason Aubrey Math 1300 Finite Mathematics
59.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
Therefore the equation of this line is
3
p − 4.8 = (q − 1.9)
2
3 3
p = q − (1.9) + 4.8
2 2
3
p = q + 1.95
2
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Jason Aubrey Math 1300 Finite Mathematics
60.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(b) Find the demand equation.
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Jason Aubrey Math 1300 Finite Mathematics
61.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(b) Find the demand equation.
Again, we want an equation of the form p = mq + b. We have
two points on the graph of the demand equation: (2.0, 4.80)
and (1.8, 5.10).
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Jason Aubrey Math 1300 Finite Mathematics
62.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(b) Find the demand equation.
Again, we want an equation of the form p = mq + b. We have
two points on the graph of the demand equation: (2.0, 4.80)
and (1.8, 5.10).
The slope of this line is:
5.10 − 4.80 0.3 3
m= =− =−
1.8 − 2.0 0.2 2
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Jason Aubrey Math 1300 Finite Mathematics
63.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(b) Find the demand equation.
Again, we want an equation of the form p = mq + b. We have
two points on the graph of the demand equation: (2.0, 4.80)
and (1.8, 5.10).
The slope of this line is:
5.10 − 4.80 0.3 3
m= =− =−
1.8 − 2.0 0.2 2
Therefore the equation of this line is
3
p − 4.80 = − (q − 2.0)
2
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Jason Aubrey Math 1300 Finite Mathematics
64.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(b) Find the demand equation.
Again, we want an equation of the form p = mq + b. We have
two points on the graph of the demand equation: (2.0, 4.80)
and (1.8, 5.10).
The slope of this line is:
5.10 − 4.80 0.3 3
m= =− =−
1.8 − 2.0 0.2 2
Therefore the equation of this line is
3
p − 4.80 = − (q − 2.0)
2
3 3
p = − q + (2.0) + 4.80
2 2
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Jason Aubrey Math 1300 Finite Mathematics
65.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(b) Find the demand equation.
Again, we want an equation of the form p = mq + b. We have
two points on the graph of the demand equation: (2.0, 4.80)
and (1.8, 5.10).
The slope of this line is:
5.10 − 4.80 0.3 3
m= =− =−
1.8 − 2.0 0.2 2
Therefore the equation of this line is
3
p − 4.80 = − (q − 2.0)
2
3 3
p = − q + (2.0) + 4.80
2 2
3
p = − q + 7.80 university-logo
2
Jason Aubrey Math 1300 Finite Mathematics
66.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(c) Find the equilibrium price and quantity.
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Jason Aubrey Math 1300 Finite Mathematics
67.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(c) Find the equilibrium price and quantity.
We set supply equal to demand and solve:
3 3
− q + 7.80 = q + 1.95
2 2
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Jason Aubrey Math 1300 Finite Mathematics
68.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(c) Find the equilibrium price and quantity.
We set supply equal to demand and solve:
3 3
− q + 7.80 = q + 1.95
2 2
3q = 5.85
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Jason Aubrey Math 1300 Finite Mathematics
69.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(c) Find the equilibrium price and quantity.
We set supply equal to demand and solve:
3 3
− q + 7.80 = q + 1.95
2 2
3q = 5.85
q = 1.95 billion bushels
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Jason Aubrey Math 1300 Finite Mathematics
70.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(c) Find the equilibrium price and quantity.
We set supply equal to demand and solve:
3 3
− q + 7.80 = q + 1.95
2 2
3q = 5.85
q = 1.95 billion bushels
Substituting, we ﬁnd p = − 3 (1.95) + 7.80 = $4.88
2
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Jason Aubrey Math 1300 Finite Mathematics
71.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
(d) Graph the two equations in the same coordinate system and
identify the equilibrium point, supply curve, and demand curve.
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Jason Aubrey Math 1300 Finite Mathematics
72.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
f
(d) Graph the two equations in the same coordinate system and
identify the equilibrium point, supply curve, and demand curve.
6
4
2
0 2 4 6 8 10 12
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Jason Aubrey Math 1300 Finite Mathematics
73.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
f
(d) Graph the two equations in the same coordinate system and
identify the equilibrium point, supply curve, and demand curve.
6
4
2
0 2 4 6 8 10 12
g
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Jason Aubrey Math 1300 Finite Mathematics
74.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
f
(d) Graph the two equations in the same coordinate system and
identify the equilibrium point, supply curve, and demand curve.
6
(1.95, 4.88)
4
2
0 2 4 6 8 10 12
g
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Jason Aubrey Math 1300 Finite Mathematics
75.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
f
(d) Graph the two equations in the same coordinate system and
identify the equilibrium point, supply curve, and demand curve.
Supply Curve
6
(1.95, 4.88)
4
2
0 2 4 6 8 10 12
g
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Jason Aubrey Math 1300 Finite Mathematics
76.
Systems in Two Variables
Graphing
Substitution
Elimination by Addition
Applications
f
(d) Graph the two equations in the same coordinate system and
identify the equilibrium point, supply curve, and demand curve.
Supply Curve
6
(1.95, 4.88)
4
2 Demand Curve
0 2 4 6 8 10 12
g
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Jason Aubrey Math 1300 Finite Mathematics
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