CHAPTER 1 LINEAR EQUATION AND FUNCTION 2 nd Semester, S.Y 2013 – 2014

BACHELOR OF ARTS IN ECONOMICS Econ 114 – MATHEMATICAL ECONOMICS

Pangasinan State UniversitySocial Science Department – PSU Lingayen

CHAPTER 1LINEAR EQUATION AND FUNCTION2nd Semester, S.Y 2013 – 2014

Origin

Y-Axis

X-Axis

QUADRANT II

(-x, y)

QUADRANT I

(x, y)

QUADRANT III

(-x, -y)

QUADRANT IV

(x, -y)

Rectangular Coordinates

Pangasinan State UniversitySocial Science Department – PSU Lingayenfall ‘ 97

Rectangular Coordinates

The X axis

Y axis

-3 -2 -1-1

(1, 2) Ordered Pair

Let’s Check Your Understanding!

1. What is the ordered pair located at Quadrant I?

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)?

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.

What is Linear Equation?

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.

Examples of Linear Equations

2x + 4y =8

6y = 3 – x

-2a + b = 5 4

Equation is in Ax + By + C = 0 form

Rewrite with both variables on left side … x + 6y =3

B =0 … x + 0 y =1

Multiply both sides of the equation by -1 … 2a – b = -5

Multiply both sides of the equation by 3 … 4x –y =-21

Examples of Nonlinear Equations

4x2 + y = 5

xy + x = 5

s/r + r = 3

The exponent is 2

There is a radical in the equation

Variables are multiplied

Variables are divided

The following equations are NOT in the standard form of Ax + By + C = 0:

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.

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.

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.

To summarize….

To find the x-intercept, plug in 0 for y.

To find the y-intercept, plug in 0 for x.

Find the x and y- intercepts of

● 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

● (0, ) is the

y-intercept

Graphing Equations

●Example: Graph the equation Solve for y first.

- 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.

Graph y = 5x + 2

Graphing Equations

●Solve for y first

4x - 3y =12 Subtract 4x from both sides

-3y = -4x + 12 Divide by -3

y = x + Simplify

y = x – 4●The equation y = x - 4 is in slope-intercept form, .

The y -intercept is -4 and the slope is . Graph the line on the coordinate plane.

Graphing Equations

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.

Undefined Slope

Positive Zero Slope

Negative

Slope of a Straight LineSlope

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.

Let’s Check Your Understanding

Compute the slope of the following diagrams.

Standard FormSlope-Intercept FormTwo-point FormIntercept FormPoint-Slope FormVertical LineHorizontal Line

Equations of a Line

Standard form for linear equations is Ax + By + C = 0

Graph the line

Write the following equations in standard form.

Standard Form

Slope Intercept Form

Slope intercept form is . 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.

Determine the Slope and the Y-intercept. Then draw a graph.

Which of the following equations represent demand curves, which represent supply curves? (Assume y represents price and x represents quantity)

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.

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).

Two-point Form – Econ Application

Price (P)Quantity

Demanded

25,000 20

20,000 40

15,000 60

10,000 80

5,000 100

Suppose that the market demand for Android tablets per week in Metro Manila is given by the following demand schedule. Derive the demand equation.

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?

Point-Slope Form

Since the slope of a nonvertical line is , the equation for point-slope form can be written

Generally most convenient for determining the equation of a straight line when one of the points on a line and its slope are given.

Point-Slope Form

Find the equation of the line passing through the point ( -1, 2) and having slope – 4.

Intercept Form

Generally most convenient for determining the equation of a straight line when its two intercepts are given. The formula is

where b is y-intercept and a is x-intercept

Intercept Form

Find the equation of the line having intercepts (0, -6) and (4, 0)

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

Vertical Lines

Find the equation of the vertical line passing through the point (5, - 4).

Vertical Lines

Find the equation of the vertical line passing through the point (5, - 4).

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

Horizontal Lines

Find the equation of the vertical line passing through the point (2, 4).

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

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 y-intercept.

Parallel Lines

Coincident Lines

Determine whether the line given by is parallel to the line given by

Perpendicular Lines

Lines that intersect at right anglesTwo 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.

Determine whether the graphs of are perpendicular

Perpendicular Lines

Conditions for Two Lines (Parallel or Perpendicular

The conditions for two lines to be parallel or perpendicular can also be stated as follows:

The Linesand

are parallel if

and perpendicular if

1. Determine whether the line given by is parallel to the line given by

2. Determine whether the line given by is parallel to the line given by

1. Determine whether the line given by is perpendicular to the line given by

2. Determine whether the line given by is perpendicular to the line given by

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.

Intersecting Lines

Find the point of intersection of the lines represented by the equation.

Intersecting Lines

Solving Solutions of a System of Equations Simultaneously

1. Substitution Method2. Comparison3. Elimination

Solution by Substitution

One algebraic method for solving systems, the substitution method, relies on havinga variable isolated.

This method is easily applied when at least one equation gives the value of one unknown in terms of the others.

Solution by Substitution

Solve the system or find the point of intersection of the lines represented by the equations: and .

Solve the system: and .

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.

Solution by Comparison

Solve the system: and .

Application – Comparison Method

The demand for umbrellas is given by equation , where P denotes the market price. The supply of umbrellas is given by equation . Find the equilibrium price and quantity

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.

Solution by Elimination

Solve the system: and . Solve the system: and . Solve the system: and .

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

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.

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.

Undefined Slope

Positive Zero Slope

Negative

𝑄𝑑=𝑎−𝑏𝑃

𝑄𝑠=𝑎+𝑏𝑃

Demand and Supply Equation

Demand Equation

Supply Equation

Where: is quantity demanded and is price of the good

is autonomous demand

is the slope or the change in price / change in

Where: is quantity supplied and is price of the good

is autonomous supply

is the slope or the change in price / change in

The demand curve for a commodity is given by the equation,, where is quantity demanded and is price. 1. What is the quantity demanded if the price is 20? 2. What is the price if the quantity demanded is 40 units?3. What is the inverse demand curve?4. 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

The supply curve for a product is represented by the equation,, where is quantity supplied and is price. 1. What is the quantity supplied if the price is Php 400? 2. What is the price if the quantity supplied is 750 units?3. What is the inverse equation of the supply curve4. What quantity would be supplied if the product is free?5. What is the highest price to be paid for this commodity?6. Graph the supply curve.

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

Elasticity Application

Flour Price

(per bag)

Quantity Demanded (per

month)

P1,000 1,400

P1,250 1,300

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, . What is the price elasticity of demand at ? At ?

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.

Price Elasticity of Demand

ElasticInelastic 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.

1. Assume that a linear demand curve is represented by the formula, . What is the price elasticity of demand at ? At ?

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

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, . What is the price elasticity of demand at ? At ?

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?

5. Because they are considered necessary for national security, 75 heavy-duty generators are bought every year, regardless of price

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.

Market Equilibrium

1. Kazuma sells cookies according to his supply function . Kioske buys cookies according to his demand function . 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 , where P denotes the market price. The supply of umbrellas is given by equation . Find the equilibrium price and quantity

Application – Market Equilibrium

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.

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.

Break-even Analysis

The cost function is given by the formula.

Revenue Function

(Price times Quantity)

Profit Function

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+7.50 𝑥

Solving for

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.

1. Find the total cost producing chairs.

2. Find the total revenue from the sale of chairs.

3. Find the total profit from the production and sale of chairs.

4. 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.

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,

3. When disposable income increases, consumption will aslo increase, but by a smaller amount. Stated mathematically, if

represents an increase in disposable income

represents the resulting change in consumption

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

Where represents consumption; is fixed, basic consumption regardless of income; is the marginal propensity to consume; and is disposable income.

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 where is aggregate consumption and 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?

CHAPTER 1 LINEAR EQUATION AND FUNCTION 2 nd Semester, S.Y 2013 – 2014

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