Chapter1 Advanced mathematics

Advanced Mathematics Chapter 1 Linear equations Advanced Mathematics Chapter 1 Linear equations Nguyen Thi Minh Tam ntmtam vnuagmail com October 20, 2020 1 1 3 Graphs of linear equations 2 1 4 Algebraic solution of simultaneous linear equations 3 1 5 Supply and demand analysis 4 1 6 Transposition of formulas 1 3 Graphs of linear equations An equation of the form dx + ey = f (1) is called a linear equation in two variables x, y d the coefficient of x e the coefficient of y The graph of a linear.

Advanced Mathematics Chapter 1: Linear equations

Nguyen Thi Minh Tamntmtam.vnua@gmail.com

October 20, 2020

1 1.3 Graphs of linear equations

2 1.4 Algebraic solution of simultaneous linear equations

3 1.5 Supply and demand analysis

4 1.6 Transposition of formulas

1.3 Graphs of linear equations

An equation of the form

Example 1 Which of the following points lie on the line3x − 5y = 25?

(5, −2), (10, 1), (5, 10)Example 2 Sketch the line: 4x + 3y = 24

Example 3 Find the point of intersection of the two lines

4x + 3y = 112x + y = 5

We say that x = 2, y = 1 is the solution of thesimultaneous linearequations

4x + 3y = 112x + y = 5

Asystem of two linear equations in two variables (a pair of

simultaneous linear equations in two variables) is of the form

ax + by = c

dx + ey = f

Graphical method for solving a system of two linear equations intwo variables

Draw the graph of the two equations on the same axes, we get twostraight lines

If the straight lines intersect at one point, the coordinates ofthis point give the solution to the system

If the straight lines are parallel, the system has no solution

If the straight lines are coincident, the system has infinitelymany solutions

When e 6= 0, the equation dx + ey = f can be transformed intothe special form y = ax + b.

a: the slopeof the line,

b: the intercept on the y axis

The slope of a straight line is the change in the value of y when xincreases by 1 unit

Example 4 Two new models of a smartphone are launched on 1January 2018 Predictions of sales are given by:

Model 1:

S1 = 4 + 0.5nModel 2:

S2 = 8 + 0.1nwhere Si (in tens of thousands) denotes the monthly sales ofmodel i after n months

a) State the values of the slope and intercept of each line andgive an interpretation

b) Illustrate the sales of both models during the first year bydrawing graphs on the same axes

c) Use the graph to find the month when sales of Model 1overtake those of Model 2

1.4 Algebraic solution of simultaneous linear equations

Drawbacks of graphical method:

It is sometimes difficult to graph accurately either or bothlines

It is often difficult to read accurately the coordinates of thepoint of intersection

This method can not be applied to solve three equations inthree variables or four equations in four variables

Example 5 Solve the system of equations

3x + 5y = 195x + 2y = −11

Elimination method for solving a system of two linear equations intwo variables

1 Add/subtract a multiple of one equation to/from a multiple ofthe other to eliminate x

2 Solve the resulting equation for y

3 Substitute the value of y into one of the original equations todeduce x

Note: We could eliminate y in step 1 and then solve the resultingequation in step 2 for y

Example 6 Solve the system of equations

3x − 2y = 4

x − 2y = 2

by eliminating one of the variables

Example 7 Solve the following systems of equations:

a) 3x − 6y = −2 b) − 5x + y = 4

Example 8 Solve the following system of equations:

1 Add/subtract multiples of the first equation to/from multiples

of the second and third equations to eliminate x

2 Add/subtract a multiple of the second equation to/from amultiple of the third to eliminate y

3 Solve the last equation for z Substitute the value of z intothe second equation to deduce y Finally, substitute thevalues of both y and z into the first equation to deduce x

1.5 Supply and demand analysis

The concept of a function

A function, f , is a rule which assigns to each incomingnumber, x , a uniquely defined outgoing number, y

x : the independent variable,

y : thedependent variable, y = f (x )

In microeconomics the quantity demanded, Q, of a gooddepends on the market price, P We might express this as

Q = f (P)Such a function is called ademand function

The demand function can be written in the form P = g (Q)

If g (Q) is a linear function, the demand function has the form

P = aQ + b, where a < 0, b > 0

Example 9 Sketch a graph of the demand function

P = −3Q + 75Hence, or otherwise, determine the value of

a) P when Q = 23

b) Q when P = 18

The supply functionis the relation between the quantity, Q,

of a good that producers plan to bring to the market and theprice, P, of the good

When the supply function is linear, it has the form

P = aQ + b, where a > 0, b > 0

In microeconomics we are concerned with the interaction ofsupply and demand.

Sketch supply and demand curves on the same diagram

At the point of intersection the market is in equilibrium.The corresponding price, P0, and quantity, Q0, are called the

equilibrium price and quantity

Example 10 The demand and supply functions of a good aregiven by

P = −2QD+ 50

P = 1

2QS+ 25where P, QD and QS denote the price, quantity demanded andquantity supplied, respectively

a) Determine the equilibrium price and quantity

b) Determine the effect on the market equilibrium if the

government decides to impose a fixed tax of $5 on each good

Suppose that there are two goods in related markets, which

we call good 1 and good 2

The demand for each good is given by

QD1 = a1+ b1P1+ c1P2

QD2 = a2+ b2P1+ c2P2where Pi and QDi denote the price and demand for the i thgood and ai, bi and ci are parameters

a1> 0, b1 < 0, a2> 0, c2 < 0The calculation of the equilibrium price and quantity in atwo-commodity market model is demonstrated in Example 11

Example 11 The demand and supply functions for twointerdependent commodities are given by

QD1 = 10 − 2P1+ P2

QD2 = 5 + 2P1− 2P2

QS1 = −3 + 2P1

QS2 = −2 + 3P2where QDi, QSi and Pi denote the quantity demanded, quantitysupplied and price of good i , respectively Determine theequilibrium price and quantity for this two-commodity model

1.6 Transposition of formulas

Mathematical modelling involves the use of formulae torepresent the relationship between economic variables.For example, the connection between price, P, and quantity,

Q, might be modelled by

P = −4Q + 100Given any value of Q it is trivial to deduce the correspondingvalue of P

Given P, it is necessary to solve an equation to deduce Q

If we are given many values of P, we should transpose theformula for P In other words, we rearrange the formula

P = an expression involving Qinto

Q = an expression involving PThe last formula enables us to find Q by replacing P by anumber

Consider the task of making Q the subject of

P = 1

3Q + 5

Example 12 Make x the subject ofa) y =r x

5

b) y = 4

2x + 1

Example 12 Make x the subject ofa) y =r x

5

b) y = 4

2x + 1

Example 12 Make x the subject ofa) y =r x

5

b) y = 4

2x + 1