PRESENTED TO PRESENTED TO

SIR. ASIF MASOODSIR. ASIF MASOOD

Allama Iqbal Open University

TOPICTOPIC

SYSTEM OF LINEAR EQUATIONS AND ITS SYSTEM OF LINEAR EQUATIONS AND ITS

APPLICATIONAPPLICATION

DISCUSSION OUTLINEDISCUSSION OUTLINEEquationsEquationsLinear equationsLinear equationsSystem of linear equationsSystem of linear equationsMathematical expression of linear equation Mathematical expression of linear equation How to develop linear equation How to develop linear equation Graphical representation of linear equationGraphical representation of linear equationPractical application / Uses of linear Practical application / Uses of linear

equationequation

EQUATION:EQUATION:DEFINATION:DEFINATION:””Equation is a symbolic statement that two Equation is a symbolic statement that two

expressions are equal”.expressions are equal”. 7 x+4=257 x+4=25

o ExpressionExpressionIt is any symbol or letter like a , b etc. It is any symbol or letter like a , b etc. o Term Term It is the sign which connect the expressionsIt is the sign which connect the expressionsMONOMINAL EXPRESSION :MONOMINAL EXPRESSION :With one term With one term MULTINOMINAL EXPRESSION:MULTINOMINAL EXPRESSION:With more than one termsWith more than one terms

TYPES OF EQUATIONSTYPES OF EQUATIONSCONDITIONALCONDITIONAL::

The equality of both sides depends upon the value of x variable . The equality of both sides depends upon the value of x variable .

The equation  2The equation  2xx – 5 = 9  – 5 = 9  is is conditionalconditional because it is only true for  because it is only true for xx = 7. Other values  = 7. Other values

of of xx do not satisfy the equation  do not satisfy the equation IDENTITYIDENTITY::

In this type x may represent any value and the two sides are In this type x may represent any value and the two sides are equal.equal.

The Equation 3x+4x=7xThe Equation 3x+4x=7x

Is an identity ,because the two sides are equal when x Is an identity ,because the two sides are equal when x represented any value.represented any value.

LINEAR EQUATIONLINEAR EQUATION DEFINATION:DEFINATION:

“ “A linear equation is any equation that A linear equation is any equation that when graphed produces a straight line and when graphed produces a straight line and the variable occurs to the first degree”. the variable occurs to the first degree”.

There are many other types of There are many other types of equations. Yet none except linear equations. Yet none except linear equations will produce a straight line when equations will produce a straight line when graphed. graphed.

3x+5=83x+5=8

Properties of linear equationProperties of linear equation

It has one or two or variables. It has one or two or variables. No variable in a linear equation is raised No variable in a linear equation is raised

to a power greater than 1.to a power greater than 1. Linear equations graph as straight lines. Linear equations graph as straight lines.

System of linear equations System of linear equations

“ “ A system of linear equations A system of linear equations means two or more means two or more

linear equations. (In plain speak: linear equations. (In plain speak: 'two or more lines') 'two or more lines')

If these two linear equations If these two linear equations intersect, that point of intersect, that point of

intersection is called the solution to intersection is called the solution to the system of linear equations “the system of linear equations “

To find the solution to systems of linear equations, To find the solution to systems of linear equations, we can use four methods:we can use four methods:

Graph Graph : by looking at where lines intersect : by looking at where lines intersect (meet) on a graph(meet) on a graph

Algebraic equation Algebraic equation : by setting the equations of : by setting the equations of the system equal to each other then solving this the system equal to each other then solving this equation.equation.

Substitution Substitution : by solving for one of the variables : by solving for one of the variables and substituting its value in to the other equation.and substituting its value in to the other equation.

EliminationElimination : Elimination involves algebraic  : Elimination involves algebraic manipulations of two or more equations. The end manipulations of two or more equations. The end goal is to eliminate a variable by creating goal is to eliminate a variable by creating opposite coefficients (The examples below should opposite coefficients (The examples below should clarify this straightforward approach).clarify this straightforward approach).

Example 1:Example 1:

A total of $12,000 is invested in two funds A total of $12,000 is invested in two funds paying 9% and 11% simple interest. If the paying 9% and 11% simple interest. If the yearly interest is $1,180, how much of yearly interest is $1,180, how much of the $12,000 is invested at each rate?the $12,000 is invested at each rate?

Before we work this problem, we must Before we work this problem, we must know the definition of simple interest. know the definition of simple interest. Simple interest can be calculated by Simple interest can be calculated by multiplying the amount invested at the multiplying the amount invested at the interest rate.interest rate.

Solution:Solution: We have two unknowns: the amount of money invested at 9% and We have two unknowns: the amount of money invested at 9% and

the amount of money invested at 11%. Our objective is to find the amount of money invested at 11%. Our objective is to find these two numbers.these two numbers.

Sentence (1) ''A total of $12,000 is invested in two funds paying Sentence (1) ''A total of $12,000 is invested in two funds paying 9% and 11% simple interest.'' can be restated as (The amount of 9% and 11% simple interest.'' can be restated as (The amount of money invested at 9%)  (The amount of money invested at money invested at 9%)  (The amount of money invested at 11%)  $12,000.11%)  $12,000.

Sentence (2) ''If the yearly interest is $1,180, how much of the Sentence (2) ''If the yearly interest is $1,180, how much of the $12,000 is invested at each rate?'' can be restated as (The $12,000 is invested at each rate?'' can be restated as (The amount of money invested at 9%)  9% + (The amount of money amount of money invested at 9%)  9% + (The amount of money invested at 11%  11%)  total interest of $1,180.invested at 11%  11%)  total interest of $1,180.

It is going to get tiresome writing the two phrases (The amount of It is going to get tiresome writing the two phrases (The amount of money invested at 9%) and (The amount of money invested at money invested at 9%) and (The amount of money invested at 11%) over and over again. So let's write them in shortcut form. 11%) over and over again. So let's write them in shortcut form. Call the phrase (The amount of money invested at 9%) by the Call the phrase (The amount of money invested at 9%) by the symbol  and call the phrase (The amount of money invested at symbol  and call the phrase (The amount of money invested at 11%) by the symbol .11%) by the symbol .

Let's rewrite sentences (1) and (2) in shortcut form.Let's rewrite sentences (1) and (2) in shortcut form.

  We have converted a narrative statement of the problem to an We have converted a narrative statement of the problem to an equivalent algebraic statement of the problem. Let's solve this equivalent algebraic statement of the problem. Let's solve this system of equations.system of equations.

A system of linear equations can be solved four different ways:A system of linear equations can be solved four different ways:   

Substitution Substitution 

EliminationElimination

Matrices Matrices 

GraphingGraphing

The Method of Substitution:The Method of Substitution: The method of substitution involves five steps:The method of substitution involves five steps: Step 1: Solve for y in equation (1).Step 1: Solve for y in equation (1).

Step 2: Substitute this value for y in equation (2). Step 2: Substitute this value for y in equation (2). This will change equation (2) to an equation with This will change equation (2) to an equation with

just one variable, x.just one variable, x.

Step 3: Solve for x in the translated equation (2).

Step 4: Substitute this value of x in the y equation you obtained in Step 1

Step 5: Check your answers by substituting the values of x and y in each of the original equations.If, after the substitution, the left side of the equation equals the right side of the equation, you know that your answers are correct.

The Method of The Method of Elimination:Elimination:

The process of elimination involves five steps:The process of elimination involves five steps:

In a two-variable problem rewrite the equations In a two-variable problem rewrite the equations so that when the equations are added, one of the so that when the equations are added, one of the variables is eliminated, and then solve for the variables is eliminated, and then solve for the remaining variable.remaining variable.

Step 1: Change equation (1) by multiplying Step 1: Change equation (1) by multiplying equation (1) by  to obtain a new and equivalent equation (1) by  to obtain a new and equivalent equation (1).equation (1).

Step 2: Add new equation (1) to equation (2) to obtain equation (3).

Step 3: Substitute  in equation (1) and solve for x.

Step 4: Check your answers in equation (2). Does

The Method of Matrices:The Method of Matrices: This method is essentially a shortcut for the This method is essentially a shortcut for the

method of elimination.method of elimination. Rewrite equations (1) and (2) without the Rewrite equations (1) and (2) without the

variables and operators. The left column contains variables and operators. The left column contains the coefficients of the x's, the middle column the coefficients of the x's, the middle column contains the coefficients of the y's, and the right contains the coefficients of the y's, and the right column contains the constants.column contains the constants.

The objective is to reorganize the original matrix into one that looks like

where a and b are the solutions to the system.

Step 1. Manipulate the matrix so that the number in cell 11 (row 1-col 1) is 1. In this case, we don't have to do anything. The number 1 is already in

the cell.

Step 2: Manipulate the matrix so that the number in cell 21 is 0. To do this we rewrite the matrix by keeping row 1 and creating a new row 2 by

adding -0.09 x row 1 to row 2.

Step 3: Manipulate the matrix so that the cell 22 is 1. Do this by multiplying row 2 by 50.

Step 4: Manipulate the matrix so that cell 12 is 0. Do this by adding

You can read the answers off the matrix as x = $7,000 and y = $5,000.

How do we “solve” a system of How do we “solve” a system of

equations??? equations???

By finding the point where two or more By finding the point where two or more equations, intersect.equations, intersect.

x + y = 6x + y = 6

y = 2xy = 2x Point of intersectionPoint of intersection

66

44

22

11 22

How do we “solve” a system of equations??? How do we “solve” a system of equations???

By finding the point where two or By finding the point where two or more equations, intersect.more equations, intersect.

x + y = 6x + y = 6

y = 2xy = 2x (2,4)(2,4)

66

44

22

22

11

The method of Graphing:The method of Graphing:

Equations of straight line Equations of straight line

Inclination of line Inclination of line The angle The angle αα (o< (o< αα <180°) measure <180°) measure

counterclockwise from positive x - axis to counterclockwise from positive x - axis to none horizontal straight line is called none horizontal straight line is called inclination of linclination of l

y l

xα

o

yyy

y

o

α =o

l װx-axis x

l

0o

y l

α =90

l װ y-axis

Slope of line Slope of line

Slope intercept form of equation Slope intercept form of equation

Y=mx+c Y=mx+c

C(0,c)P(x,y)

x

y l

o

cα

Positive SlopePositive Slope When a line slopes up from left to right, it has a When a line slopes up from left to right, it has a

positive slope. This means that a positive change in positive slope. This means that a positive change in yy is associated with a positive change in is associated with a positive change in xx. The steeper . The steeper the slope, the greater the rate of change in the slope, the greater the rate of change in yy in in relation to the change in relation to the change in xx. .

Negative SlopeNegative Slope

When a line slopes down from left to right, it has When a line slopes down from left to right, it has a negative slope. This means that a negative a negative slope. This means that a negative change in change in yy is associated with a positive change is associated with a positive change

in in xx..

Positive Slope form of equation Positive Slope form of equation

Y=mx+c Y=mx+c

C(0,c)P(x,y)

x

y l

o

cα

Systems of linear equations can be usedSystems of linear equations can be used

Systems of linear equations can be Systems of linear equations can be used in various subject areas and are used in various subject areas and are used to solve for unknowns. With n used to solve for unknowns. With n equations, the highest possible equations, the highest possible number of unknowns that can be number of unknowns that can be solved for is n. Complications occur solved for is n. Complications occur when the number of unknowns is when the number of unknowns is unequal to the number of equations unequal to the number of equations in the system.in the system.

Systems of linear equations can be usedSystems of linear equations can be used

EconomicsEconomics Linear Equations can be used to find out the best Linear Equations can be used to find out the best

method of maximizing profits.method of maximizing profits.

BiologyBiology Systems of linear equations can also be used in Systems of linear equations can also be used in

biology.biology.

ChemistryChemistry Systems of linear equations can also be used in Systems of linear equations can also be used in

chemistry to balance chemical equations.chemistry to balance chemical equations.

PhysicsPhysics Systems of linear equations and matrices can be Systems of linear equations and matrices can be

used to find missing currents with given voltages.used to find missing currents with given voltages.