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CHAPTER 1
LINEAR EQUATION AND FUNCTION
2nd Semester, S.Y 2013 – 2014
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Rectangular Coordinates
Y-Axis
QUADRANT II
QUADRANT I
(-x, y)
(x, y)
Origin
QUADRANT III
(-x, -y)
QUADRANT IV
X-Axis
(x, -y)
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Rectangular Coordinates
Y axis
3
2
(1, 2) Ordered Pair
1
-3
-2
The X axis
-1
1
2
3
x
-1
-2
-3
Pangasinan State University
Social Science Department – PSU Lingayen
fall ‘ 97
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Let’s Check Your Understanding!
1. What is the ordered
located at Quadrant I?
pair
2. What is the order pair plotted
in Quadrant IV?
3. What is the point of origin?
4. What is x-coordinate in the
ordered pair (-5, -3)?
5. What is the value of y or
ordinate in the ordered pair (4, 5)?
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
What is Linear Equation?
 A linear equation is an equation whose graph
or solutions form a straight line on a
coordinate plane.
 Real life economic situations of linear
equations include demand and supply
analysis, cost and revenue, consumption and
savings, production, stock exchange, etc.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Identifying a Linear Equation
●
●
●
●
●
●
●
Ax + By + C = 0
A, B and C are constants.
The are two variables (x and y)
The variables are added or subtracted.
At least one of A and B is nonzero
Besides x and y, other commonly used variables
are m and n, a and b, and r and s.
There are no radicals in the equation.
Every linear equation graphs as a straight line.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Examples of Linear Equations
2x + 4y =8
Equation is in Ax + By + C = 0 form
6y = 3 – x
Rewrite with both variables
on left side … x + 6y =3
x=1
B =0 … x + 0 y =1
-2a + b = 5
Multiply both sides of the equation by -1
… 2a – b = -5
4x  y
 7
3
Multiply both sides of the equation by 3
… 4x –y =-21
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Examples of Nonlinear Equations
The following equations are NOT in the
standard form of Ax + By + C = 0:
4x2 + y = 5
x4
xy + x = 5
s/r + r = 3
The exponent is 2
There is a radical in the equation
Variables are multiplied
Variables are divided
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
X and Y – Intercepts
 The x-intercept is the point where a line crosses the
x-axis.
The general form of the x-intercept is (x, 0).
The
y-coordinate will always be zero.
 The y-intercept is the point where a line crosses the
y-axis.
The general form of the y-intercept is (0, y).
The
x-coordinate will always be zero.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Finding the x-intercept
● For the equation 2x + y = 6, we know that
y must equal 0. What must x equal?
● Plug in 0 for y and simplify.
2x + 0 = 6
2x = 6
x=3
● So (3, 0) is the x-intercept of the line.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Finding the y-intercept
 For the equation 2x + y = 6, we know that
x must equal 0. What must y equal?
● Plug in 0 for x and simplify.
2(0) + y = 6
0+y=6
y=6
● So (0, 6) is the y-intercept of the line.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
To summarize….
 To find the x-intercept, plug in 0 for y.
 To find the y-intercept, plug in 0 for x.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Find the x and y- intercepts of 𝑥 = 4𝑦 – 5
● x-intercept:
● Plug in y = 0
x = 4y - 5
x = 4(0) - 5
x=0-5
x = -5
● (-5, 0) is the
x-intercept
● y-intercept:
● Plug in x = 0
x = 4y - 5
0 = 4y - 5
5 = 4y
5
=y
4
●
5
(0, 4 )
is the
y-intercept
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Graphing Equations
● Example: Graph the equation −5𝑥 + 𝑦 = 2
Solve for y first.
-5𝑥 + 𝑦 = 2 Add 5x to both sides
y = 5x + 2
● The equation y = 5x + 2 is in slope-intercept form,
y = mx+b. The y-intercept is 2 and the slope is 5.
Graph the line on the coordinate plane.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Graphing Equations
Graph y = 5x + 2
x
y
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Graphing Equations
Graph 4𝑥 − 3𝑦 = 12
● Solve for y first
4x - 3y =12
Subtract 4x from both sides
-3y = -4x + 12 Divide by -3
y
-4
= -3
12
x + -3
Simplify
4
y = 3x – 4
● The equation y = - 4 is in slope-intercept form,
4
𝑦 = 𝑚𝑥 + 𝑏. The y -intercept is -4 and the slope is 3
. Graph the line on the coordinate plane.
4
3x
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Slope
 Slope is the ratio of vertical change to the
horizontal change (rise/run) of a line.
 Slope in a linear equation shows if the line is
ascending (positive) or descending (negative).
 Slope can also show the rate of change.
 The letter m is used to represent slope in a
formula.
Y
rise
m

X
run
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Slope of a Straight Line
Negative
Undefined Slope
Positive
Zero Slope
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Let’s Check Your Understanding
Study the four accompanying diagrams. Consider the following statements and indicate
which diagram matches each statement. Which variable would appear on the horizontal and
which on the vertical axis? In each of these statements, is the slope positive, negative, zero,
or infinity?
a. If the price of movies increases, fewer consumers go to see movies.
b. More experienced workers typically have higher incomes than less experienced workers.
c. Whatever the temperature outside, people consume the same number of hotdogs per day.
d. Consumers buy more frozen yogurt when the price of ice cream goes up.
e. Research finds no relationship between the number of diet books purchased and the
number of pounds lost by the average dieter.
f. Regardless of its price, consumers in Lingayen buy the same quantity of salt.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Let’s Check Your Understanding
Compute the slope of the following diagrams.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Equations of a Line
Standard Form
Slope-Intercept Form
Two-point Form
Intercept Form
Point-Slope Form
Vertical Line
Horizontal Line
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Standard Form
Standard form for linear equations is Ax + By + C = 0
Graph the line 4𝑥 + 5𝑦 − 20 = 0
Write the following equations in standard form.
A. 𝟐 = 𝟒𝒙 − 𝒚
B. 𝒙 = 𝒚 − 𝟏
C. 𝒚 = 𝟐𝒙 + 𝟑
D. 𝒚 = −𝟒𝒙 − 𝟓
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Slope Intercept Form
 Slope intercept form is 𝑦 = 𝑚𝑥 + 𝑏.
 This form makes it easy to find the slope (m)
and the y-intercept (b).
 Working with this form is simple, so it is used
more often than other forms.
Example: y= ¾x + 3
* ¾ is the slope.
* 3 is the point where the line crosses the Y-axis.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Let’s Check Your Understanding!
Determine the Slope and the Y-intercept. Then draw
a graph.
1. 4𝑥 − 𝑦 − 1 = 0
2. 𝑦 = 3𝑥 − 5
5
2
3. 𝑦 = 𝑥 − 3
4. 2𝑥 + 3𝑦 − 6 = 0
5. 𝑦 = 2𝑥 + 1
Pangasinan State University
Social Science Department – PSU Lingayen
fall ‘ 97
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Let’s Check Your Understanding!
Which of the following equations represent demand
curves, which represent supply curves? (Assume y
represents price and x represents quantity)
1.
2.
3.
4.
5.
𝑥 − 2𝑦 = 0
3𝑥 + 4𝑦 − 10 = 0
2𝑥 − 3𝑦 + 1 = 0
3𝑥 + 4𝑦 − 12 = 0
5𝑥 − 𝑦 − 10 = 0
Pangasinan State University
Social Science Department – PSU Lingayen
fall ‘ 97
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Two-point Form
 The two-point form for a straight line, is
generally most convenient for determining the
equation of a straight line when two of its points
are given.
 𝑦 − 𝑦1 =
𝑦2−𝑦1
𝑥2−𝑥1
(𝑥 − 𝑥1)
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Two-point Form
Find the equation of the line passing through the
points (2, 3) and (3, 7).
Find the equation of the line passing through the
points (3, 4) and (-5, 2).
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Two-point Form – Econ Application
Suppose that the market demand for Android tablets per
week in Metro Manila is given by the following demand
schedule. Derive the demand equation.
Price (P)
Quantity
Demanded
25,000
20,000
15,000
10,000
5,000
20
40
60
80
100
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Two-point Form –Econ Application
When the price is P500, 50 MP3 players of a
fixed type are available for sale; when the
price is P750, 100 of the MP3 players are
available. What is the supply equation?
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Point-Slope Form
 Since the slope of a nonvertical line is 𝑚 =
𝑦2−𝑦1
, the equation for point-slope form can
𝑥2−𝑥1
be written
𝑦 − 𝑦1 = 𝑚 (𝑥 − 𝑥1)
 Generally most convenient for determining the
equation of a straight line when one of the
points on a line and its slope are given.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Point-Slope Form
Find the equation of the line passing through
the point ( -1, 2) and having slope – 4.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Intercept Form
 Generally most convenient for determining the
equation of a straight line when its two
intercepts are given. The formula is
𝑏
𝑦 =− 𝑥+𝑏
𝑎
where b is y-intercept and a is x-intercept
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Intercept Form
 Find the equation of the line having intercepts
(0, -6) and (4, 0)
𝑏 = −6
𝑎=4
𝑏
𝑦 =− 𝑥+𝑏
𝑎
−6
𝑦=−
x + −6
4
3
2
𝑦 = x − 6 or 3𝑥 − 2𝑦 − 12 = 0
Pangasinan State University
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Vertical Lines
 Since the slope of a vertical line is undefined,
the above formulas are not appropriate for
obtaining the equations of vertical lines.
 A vertical line passing through the point (x1, y1)
has the equation 𝑥 = 𝑥1
Pangasinan State University
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Vertical Lines
 Find the equation of the vertical line passing
through the point (5, - 4).
𝑥 = 𝑥1 = 5
𝑥=5
Pangasinan State University
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Vertical Lines
 Find the equation of the vertical line passing
through the point (5, - 4).
𝑥 = 𝑥1 = 5
𝑥=5
Pangasinan State University
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Horizontal Lines
 Since a horizontal line has zero slope, its
equation maybe obtained from the two-point,
point-slope, or slope-intercept forms.
 A horizontal line passing through the point (x1,
y1) has the equation 𝑦 = 𝑥1
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Horizontal Lines
 Find the equation of the vertical line passing
through the point (2, 4).
𝑦 = 𝑦1 = 5
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Parallel, Perpendicular and Intersecting
Lines
A system of linear equations consists of two or
more linear equations with the same variables.
 Any two lines in a plane are either parallel or
intersecting lines.
 Lines that intersect at right angles re
perpendicular
Pangasinan State University
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Parallel and Coincident Lines
 Parallel lines are two lines in a plane that have
equal angles of inclination and therefore have
equal slopes and conversely.
 Two lines with different -intercepts are parallel if
they have the same slope.
 Also, two vertical lines are parallel.
Coincident (equivalent equations) are two lines
having the same slope and the same yintercept.
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Parallel Lines
2𝑥 − 𝑦 − 1 = 0
2𝑥 − 𝑦 + 5 = 0
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Econ 114 – MATHEMATICAL ECONOMICS
Coincident Lines
2𝑥 − 𝑦 − 1 = 0
4𝑥 − 2𝑦 − 2 = 0
Pangasinan State University
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Let’s Check Your Understanding
 Determine whether the line given by 𝑦 = −3𝑥 + 5
is parallel to the line given by 6𝑥 + 2𝑦 − 1 = 0
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Perpendicular Lines
 Lines that intersect at right angles
Two lines that are perpendicular have slopes
which are the negative reciprocals of each
other, and conversely.
 Two lines are perpendicular if the product of their
slopes is -1 or if one line is vertical and the other
is horizontal.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Perpendicular Lines
 Determine whether the graphs of 2𝑥 + 𝑦 − 8 =
1
0 𝑎𝑛𝑑 𝑦 = 𝑥 + 7 are perpendicular
2
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Perpendicular Lines
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Econ 114 – MATHEMATICAL ECONOMICS
Conditions for Two Lines (Parallel or
Perpendicular
The conditions for two lines to be parallel or
perpendicular can also be stated as follows:
 The Lines 𝐴1 𝑥 + 𝐵1 𝑦 + 𝐶1 = 0 and 𝐴2 𝑥 + 𝐵2 𝑦 +
𝐶2 = 0
are parallel if 𝑨𝟏 𝑩𝟐 − 𝑨𝟐 𝑩𝟏 = 𝟎
and perpendicular if 𝑨𝟏 𝑨𝟐 + 𝑩𝟏 𝑩𝟐 = 𝟎
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Conditions for Two Lines (Parallel or
Perpendicular
1. Determine whether the line given by 2𝑥 +
6𝑦 − 4 = 0 is parallel to the line given by
𝑥 + 3𝑦 − 9 = 0
2. Determine whether the line given by 2𝑥 +
6𝑦 − 4 = 0 is parallel to the line given by 4𝑥 +
12𝑦 − 8 = 0
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Conditions for Two Lines (Parallel or
Perpendicular
1. Determine whether the line given by
2𝑥 + 6𝑦 − 4 = 0 is perpendicular to
the line given by 3𝑥 − 𝑦 + 4 = 0
2. Determine whether the line given by
2𝑥 + 6𝑦 − 4 = 0 is perpendicular to the
line given by 2𝑥 + 𝑦 − 4 = 0
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Intersecting Lines
 Two straight nonparallel lines that intersect in
exactly one point.
 The point of intersection of two lines can be
found
by
solving
their
equations
simultaneously.
 Two lines which intersect corresponds to the
algebraic condition that their equations are
independent and consistent, and therefore have
a simultaneous solution.
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Intersecting Lines
 Find the point of intersection of the lines
represented by the equation.
y − 𝑥 − 1 = 0 and y + 𝑥 − 3 = 0 .
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Intersecting Lines
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Econ 114 – MATHEMATICAL ECONOMICS
Solving Solutions of a System of
Equations Simultaneously
1. Substitution Method
2. Comparison
3. Elimination
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Solution by Substitution
One algebraic method for solving systems,
the substitution method, relies on having
a variable isolated.
This method is easily applied when at least
one equation gives the value of one
unknown in terms of the others.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Solution by Substitution
 Solve the system or find the point of
intersection of the lines represented by the
equations: 𝑥 + 𝑦 − 7 = 0 and 3𝑥 − 𝑦 − 1 = 0 .
 Solve the system: 3𝑥 + 2𝑦 − 11 = 0 and 𝑦 =
4𝑥 .
 Solve the system: 5𝑥 + 3𝑦 − 3 = 0 and 4𝑥 +
𝑦−8=0.
Pangasinan State University
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BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Solution by Comparison
This method of solving systems of linear
equations uses the transitive property of equality,
which states that quantities equal to the same
quantity are equal. This is used when the same
unknown in the two equations has coefficient 1.
The value of the other unknown can then be
found.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Solution by Comparison
 Solve the system: 𝑥 + 2𝑦 − 4 = 0 and 𝑥 − 3𝑦 +
1=0.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Application – Comparison Method
The demand for umbrellas is given by equation
𝑄𝑑 = 800 − 2𝑃, where P denotes the market price.
The supply of umbrellas is given by equation 𝑄𝑠 =
500 + 𝑃. Find the equilibrium price and quantity
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Solution by Elimination
The elimination method for solving systems of
equations makes use of the addition principle.
This states that when two quantities are equal,
adding the same number to each quantity results
in equal sums.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Solution by Elimination
 Solve the system: 𝑥 + 𝑦 − 8 = 0 and 𝑥 − 𝑦 −
2=0.
 Solve the system: 𝑥 + 𝑦 − 2 = 0 and 2𝑥 − 𝑦 −
4=0.
 Solve the system: 2𝑥 − 3𝑦 − 2 = 0 and 𝑥 + 𝑦 −
6=0.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Application of Straight Lines in
Economics






Demand and Supply Curves
Market Equilibrium
Price Elasticity of Demand
Break-even Analysis
Consumption Function and Income Function
Profit Maximization
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Demand and Supply Curves
Quantity demanded and quantity supplied of a
given commodity are functions of a number of
variables including price of the commodity, prices
of substitutes and complements, income, tastes,
habits and so forth.
There is an inverse relationship exists between
price and quantity demanded and a direct
relationship between price and quantity supplied.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Demand and Supply Curves
Quantity demanded and quantity supplied of a
given commodity are functions of a number of
variables including price of the commodity, prices
of substitutes and complements, income, tastes,
habits and so forth.
There is an inverse relationship exists between
price and quantity demanded and a direct
relationship between price and quantity supplied.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Demand and Supply Curves
Negative
Undefined Slope
Positive
Zero Slope
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Demand and Supply Equation
𝑄𝑑 = 𝑎 − 𝑏𝑃
Demand Equation
Where: 𝑄𝑑 is quantity demanded and 𝑃 is price of the good
𝑎 is autonomous demand
𝑏 is the slope or the change in price / change in 𝑄𝑑
𝑄𝑠 = 𝑎 + 𝑏𝑃
Supply Equation
Where: 𝑄𝑠 is quantity supplied and 𝑃 is price of the good
𝑎 is autonomous supply
𝑏 is the slope or the change in price / change in 𝑄𝑠
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Let’s Check Your Understanding
The demand curve for a commodity is given by the
equation, 𝑄𝑑 = 200 − 5𝑃, where 𝑄𝑑 is quantity demanded
and 𝑃 is price.
1.
2.
3.
4.
What is the quantity demanded if the price is 20?
What is the price if the quantity demanded is 40 units?
What is the inverse demand curve?
What quantity would be demanded if the commodity
were free?
5. What is the highest price that would be paid for this
commodity?
6. Graph the demand curve
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Let’s Check Your Understanding
The supply curve for a product is represented by the
1
equation, 𝑄𝑠 = 600 + 𝑃, where 𝑄𝑠 is quantity supplied and
2
𝑃 is price.
1.
2.
3.
4.
5.
6.
What is the quantity supplied if the price is Php 400?
What is the price if the quantity supplied is 750 units?
What is the inverse equation of the supply curve
What quantity would be supplied if the product is free?
What is the highest price to be paid for this commodity?
Graph the supply curve.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Elasticity Analysis
Elasticity is a measure of the responsiveness
of one variable to another. The greater the
elasticity, the greater the responsiveness.
Four Types




Price Elasticity of Demand
Income Elasticity of Demand
Cross Price Elasticity
Price Elasticity of Supply
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Elasticity Application
 When the price of pandesal per piece is P2, Santos family usually bought 20
pieces every morning. When the price increased to P3.00, the family now
buys 18 pieces. Based on the data, compute for the price elasticity of
demand. Interpret the computed elasticity value and determine whether the
good (pandesal) is a necessity or a luxury.
 Assume that a linear demand curve is represented by the formula, 𝑄𝑑 =
400 − 2𝑃. What is the price elasticity of demand at 𝑃 = 30? At 𝑃 = 10?
 Given the demand data of Sunrise bakery for loaf bread subject to price
increase of flour, calculate the cross elasticity for loaf bread. Identify if it is
elastic, unitary or inelastic and determine whether the loaf bread and flour
are substitutes or complements.
Flour
Price
(per bag)
Quantity
Demanded (per
month)
P1,000
1,400
P1,250
1,300
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Price Elasticity of Demand
Elastic
Inelastic
Unitary
 Perfectly elastic and Perfectly inelastic
 Demand is ELASTIC when the numerical value of
elasticity is greater than 1.
 Demand is INELASTIC when the numerical value of
elasticity is less than 1.
 Demand is UNIT ELASTIC when the numerical value
of elasticity equals 1.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Let’s Check Your Understanding
1.
Assume that a linear demand curve is represented by the formula, 𝑄𝑑 =
500 − 2𝑃. What is the price elasticity of demand at 𝑃 = 30? At 𝑃 = 10?
2.
The demand data of Tomas for chicken inasal subject to price increase of
pork barbecue is described by the table below. Solve for the cross-price
elasticity of demand. Identify if it is elastic, unitary or inelastic and
determine whether chicken inasal and pork barbecue are substitutes or
complements. (3 points)
Pork barbecue
( price per stick)
Quantity
Demanded (per
month)
P20
14
P30
17
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Let’s Check Your Understanding
1.
A firm has analyzed its sales and found that its customers will buy 20%
more units of its products with each P5 reduction in the unit price. When
the price is P25, the firms sells P600 units. What is the equation of the
demand function for this product?
2.
Assume that a linear demand curve is represented by the formula, 𝑄𝑑 =
500 − 2𝑃. What is the price elasticity of demand at 𝑃 = 30? At 𝑃 = 10?
3.
At a price P1, 200 per unit, a firm will supply 9,000 of processor chips for
tablets; at P900 per unit it will supply 6,000 units. Determine the equation
of the supply function for this product.
4.
According to the terms of the contract between Company A and the
internet provider company, Company A pays the internet provider company
P1500 per month for unlimited internet connection. What is the supply
equation?
Because they are considered necessary for national security, 75 heavyduty generators are bought every year, regardless of price
5.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Market Equilibrium
Market equilibrium is said to occur at that point (price) at
which the quantity of a commodity demanded is equal to
the quantity supplied.
Algebraically, the equilibrium quantity and equilibrium price
are found by solving the demand and supply equations
simultaneously.
In general, for an equilibrium to be meaningful, the value of
x and y must be positive or zero.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Market Equilibrium
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Application – Market Equilibrium
1. Kazuma sells cookies according to his supply
function 𝑄𝑠 = −15 + 3𝑃. Kioske buys cookies
according to his demand function 𝑄𝑑 = 27 − 4𝑃
. If the two of them make up the entire market for
cookies, what will the equilibrium price and
quantity (in thousands) be?
2. The demand for umbrellas is given by equation
𝑄𝑑 = 800 − 2𝑃 , where P denotes the market
price. The supply of umbrellas is given by
equation 𝑄𝑠 = 500 + 𝑃. Find the equilibrium price
and quantity
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Break-even Analysis
In economics & business, the break-even point
(BEP) is the point at which cost or expenses and
revenue are equal: there is no net loss or gain, and
one has "broken even". For example, if a business
firm sells fewer than 200 computers each month,
it will make a loss, if it sells more, it will be a profit.
With this information, the business managers will
then need to see if they expect to be able to make
and sell 200 computers per month.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Break-even Analysis
The money that a business spends to manufacture a product is its cost.
The total cost of production can be thought of as a function , where is the
cost of producing units. When the company sells the product, it takes in
money. This is revenue and can be thought of as a function , where is the
total revenue from the sale of units. Total profit is the money taken in less
the money spent, or total revenue minus total cost. Total profit from the
production and sale of units is a function given by Profit Revenue Cost If is
greater than , there is a gain and is positive. If is greater than , there is a
loss and is negative. When the company breaks even.
There are two kinds of costs. First, there are costs like rent, insurance,
machinery, and so on. These costs, which must be paid regardless of how
many items are produced, are called fixed costs. Second, costs for labor,
materials, marketing, and so on are called variable costs, because they
vary according to the amount being produced. The sum of the fixed cost
and the variable cost gives the total cost.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Break-even Analysis
The cost function is given by the formula.
C (𝑥) = 𝑎 + 𝑏𝑥.
Revenue Function
𝑅(𝑥) = 𝑃𝑥 (Price times Quantity)
Profit Function
𝑃 𝑥 = 𝑅 𝑥 − 𝐶(𝑥)
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Break-even Analysis
Suppose the fixed cost of production for a commodity is P5,000, the
variable cost is P7.50 per unit and the commodity sells for P10 per unit.
What is the break-even quantity?
𝐶(𝑥) = 5000 + 7.50𝑥
𝑅 𝑥 = 10𝑥
Solving for 𝑥,
2.5𝑥 = 5000
𝑥 = 2,000
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Break-even Analysis
V Designs is planning to make a new chair. Fixed costs will be P90, 000
and it will cost P150 to produce each chair (Variable costs). Each chair sells
for P400.
Find the total cost 𝐶 (𝑥) producing 𝑥 chairs.
Find the total revenue 𝑅 𝑥 from the sale of 𝑥 chairs.
Find the total profit 𝑃 (𝑥) from the production and sale of 𝑥 chairs.
What profit will be the company realize from the production and sale of
300 chairs? Of 800 chairs?
5. Determine the break-even point.
6. Graph the total cost, total revenue, and total profit function using the
same set of axes.
1.
2.
3.
4.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Consumption Function
National income analysis provides an interesting example of the use of linear
relations, since the consumption function is frequently assumed to be linear over
relatively short ranges or “in the short run”. The analysis, in one of its simplest
froms, is based on the following assumptions.
1. There is some absoulute amount of consumption necessary to maintain life,
even thogh there will be no money income.
2. Consumption is related to disposable income, that is, C = 𝑓(𝑌)
3. When disposable income increases, consumption will aslo increase, but by a
smaller amount. Stated mathematically, if
𝑦𝑑 represents an increase in disposable income
and
𝑐 represents the resulting change in consumption
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Consumption Function
4. The proportion of an increment in disposable income which will
be consumed is constant. This proportion is referred to as the
“marginal propensity to consume” .
These assumptions can be translated into the point-slope form of
the equation for a straight line
𝑐 = 𝑎 + 𝑏𝑦𝑑
Where 𝑐 represents consumption; 𝑎 is fixed, basic consumption
regardless of income; 𝑏 is the marginal propensity to consume;
and 𝑦𝑑 is disposable income.
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS
Consumption Function
1. When national disposable income is 0, national consumption is 5
(in trillions of pesos). For the economy as a whole consumption is
linearly related to national disposable income as follows. At each
level of disposable income, consumption equals 5 (in trillion of
pesos) plus 80% of disposable income. What is the equation that
expresses this relationship? What is aggregate consumption when
disposable income is 40 (in trillion of pesos)
2. Suppose that the consumption function is given by 𝐶 = 10 +
0.55𝑌, where 𝐶 is aggregate consumption and Y is disposable
income (in trillion of pesos). What is aggregate consumption when
disposable income is 10 trillion of pesos? When disposable income
is 5 billion dollars, what proportion of aggregate consumption is
consumption of disposable income?
Pangasinan State University
Social Science Department – PSU Lingayen
BACHELOR OF ARTS IN ECONOMICS
Econ 114 – MATHEMATICAL ECONOMICS