Linear Inequalities and Linear Equations

8/11/2019 Linear Inequalities and Linear Equations

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Linear Inequalities and Linear Equations

Inequalities

The term inequality is applied to any statement involving one of the symbols , , .

Example of inequalities are:

i. x 1

ii. x + y + 2z > 16

iii. p2+ q2 1/2

iv. a2+ ab > 1

Fundamental Properties of Inequalities

1. If a b and c is any real number, then a + c b + c.

For example, -3 -1 implies -3+4 -1 + 4.

2. If a b and c is positive, then ac bc.

For example, 2 3 implies 2(4) 3(4).

3. If a b and c is negative, then ac bc.

For example, 3 9 implies 3(-2) 9(-2).

4. If a b and b c, then a c.

For example, -1/2 2 and 2 8/3 imply -1/2 8/3.


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Solution of Inequality

By solution of the one variable inequality2x + 3 7 we mean any number which

substituted for x yields a true statement.

For example, 1 is a solution of 2x + 3 7 since 2(1) + 3 = 5 and 5 is less than and equal to

7.

By a solution of the two variable inequalityx - y 5 we mean any ordered pair of numbers

which when substituted for x and y, respectively, yields a true statement.

For example, (2, 1) is a solution of x - y 5 because 2-1 = 1 and 1 5.

By a solution of the three variable inequality2x - y + z 3 we means an ordered triple of

number which when substituted for x, y and z respectively, yields a true statement.

For example, (2, 0, 1) is a solution of 2x - y + z 3.

A solution of an inequality is said to satisfy the inequality. For example, (2, 1) is satisfy x -

y 5.

Two or more inequalities, each with the same variables, considered as a unit, are said to

form a system of inequalities. For example,

x 0

y 0

2x + y 4

Notethat the notion of a system of inequalities is analogous to that of a solution of asystem of equations.

Any solution common to all of the inequalities of a system of inequalities is said to be a

solution of that system of inequalities. A system of inequalities, each of whose members is

linear, is said to be a system of linear inequalities.

Geometric Interpretation of Inequalities

An inequality in two variable x and y describes a region in the x-y plane (called its graph),


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namely, the set of all points whose coordinates satisfy the inequality.

The y-axisdivide, the xy-plane into two regions, called half-planes.

Right half-plane

The region of points whose coordinates satisfy inequality x > 0.

Left half-plane

The region of points whose coordinates satisfy inequality x < 0.

Similarly, the x-axisdivides the xy-plane into two half-planes.

Upper half-plane

In which inequality y > 0is true.

Lower half-plane

In which inequality y < 0is true.

What is x-axis and y-axis? They are simply lines. So, the above arguments can be applied

to any line.

Every line ax + by = c divides the xy-plane into two regions called its half-planes.

On one half-plane ax + by > c is true.

On the other half-plane ax + by < c is true.

Linear Equations

One Unknown

A linear equation in one unknown can always be stated into the standard form

ax = b

where x is an unknown and a and b are constants. If a is not equal to zero, this equation

has a unique solution

x = b/a

Two Unknowns

A linear equation in two unknown, x and y, can be put into the form


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ax + by = c

where x and y are two unknowns and a, b, c are real numbers. Also, we assume that a

and b are no zero.

Solution of Linear Equation

A solution of the equation consists of a pair of number, u = (k1, k2), which satisfies the

equation ax + by = c. Mathematically speaking, a solution consists of u = (k1, k2)

such that ak1+ bk2= c. Solution of the equation can be found by assigning arbitrary

values to x and solving for y ORassigning arbitrary values to y and solving for x.

Geometrically, any solution u = (k1, k2)of the linear equation ax + by = cdetermine a

point in the cartesian plane. Since a and b are not zero, the solution u correspond precisely

to the points on a straight line.

Two Equations in the Two Unknowns

A system of two linear equations in the two unknowns x and y is

a1x + b1x = c1a2x + b2x = c2

Where a1, a2, b1, b2 are not zero. A pair of numbers which satisfies both equations is called

a simultaneous solution of the given equations or a solution of the system of equations.

Geometrically, there are three casesof a simultaneous solution

1. If the system has exactly one solution, the graph of the linear equations intersect in

one point.

2. If the system has no solutions, the graphs of the linear equations are parallel.

3. If the system has an infinite number of solutions, the graphs of the linear equations

coincide.

The special cases (2) and (3) can only occur when the coefficient of x and y in the two


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linear equations are proportional.

OR => a1b2-a2b1= 0 => = 0

The system has no solution when

The solution to system

a1x + b1x = c1a2x + b2x = c2

can be obtained by the elimination process, whereby reduce the system to a single equation

in only one unknown. This is accomplished by the following algorithm

ALGORITHM

Step 1 Multiply the two equation by two numbers which are such that

the resulting coefficients of one of the unknown are negative of

each other.

Step 2 Add the equations obtained in Step 1.

The output of this algorithm is a linear equation in one unknown. This equation may besolved for that unknown, and the solution may be substituted in one of the original

equations yielding the value of the other unknown.

As an example, consider the following system

3x + 2y = 8 ------------ (1)

2x - 5y = -1 ------------ (2)

Step 1: Multiply equation (1) by 2 and equation (2) by -3


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6x + 4y = 16

-6x + 15y = 3

Step 2: Add equations, output of Step 1

19y = 19

Thus, we obtain an equation involving only unknown y. we solve for y to obtain

y = 1

Next, we substitute y =1 in equation (1) to get

x = 2

Therefore, x = 2 and y = 1 is the unique solutionto the system.

n Equations in n Unknowns

Now, consider a system of n linear equations in n unknowns

a11x1+ a12x2+ . . . + a1nxn= b1a21x1+ a22x2+ . . . + a2nxn= b2

. . . . . . . . . . . . . . . . . . . . . . . . .

an1x1+ an2x2+ . . . + annxn= bn

Where the aij, biare real numbers. The number aijis called the coefficient of xjin the i

th

equation, and the number biis called the constant of the ith equation. A list of values for

the unknowns,

x1= k1, x2= k2, . . . , xn= kn

or equivalently, a list of n numbers

u = (k1, k2, . . . , kn)

is called a solution of the system if, with kj substituted for xj, the left hand side of each

equation in fact equals the right hand side.


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The above system is equivalent to the matrix equation.

or, simply we can write A = B, where A = (a ij), = (xi), and B = (bi).

The matrix is called the coefficient matrixof the system of n linear

equations in the system of n unknown.

The matrix is called the augmented matrixof n linear

equations in n unknown.

Note for algorithmic nerds: we store a system in the computer as its augmented matrix.

Specifically, system is stored in computer as an N (N+1) matrix array A, the augmented

matrix array A, the augmented matrix of the system. Therefore, the constants b1, b2, . . . ,

bnare respectively stored as A1,N+1, A2,N+1, . . . , AN,N+1.

Solution of a Triangular System

If aij= 0 for i > j, then system of n linear equations in n unknown assumes the triangular

form.

a11x1+ a12x2+ . . . + a1,n-1xn-1 + a1nxn = b1 a22x2+ . . . + a2,n-1xn-1 + a2nxn = b2


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. . . . . . . . . . . . . . . . . . . . . . . . . . . .

an-2,n-2xn-2+ an-2,n-1xn-1 + an-2,nxn-1 + a2nxn= b2an-1,n-1xn-1+ an-1,nxn= bn-1

amnxn= bn

Where |A| = a11a22. . . ann; If none of the diagonal entries a11,a22, . . ., anniszero, the system has a unique solution.

Back Substitution Method

we obtain the solution of a triangular system by the technique of back substitution,

consider the above general triangular system.

1. First, we solve the last equation for the last unknown, xn;

xn= bn/ann

2. Second, we substitute the value of xnin the next-to-last equation and solve it for the

next-to-last unknown, xn-1:

.

3. Third, we substitute these values for xnand xn-1in the third-from-last equation and solve

it for the third-from-last unknown, xn-2:

.

In general, we determine xkby substituting the previously obtained values of xn, xn-1, . . . ,

xk+1in the kthequation.

.


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Gaussian Elimination

Gaussian elimination is a method used for finding the solution of a system of linear

equations. This method consider of two parts.

1. This part consists of step-by-step putting the system into triangular system.

2. This part consists of solving the triangular system by back substitution.

x - 3y - 2z = 6 --- (1)

2x - 4y + 2z = 18 --- (2)

-3x + 8y + 9z = -9 --- (3)

First Part

Eliminate first unknown x from the equations 2 and 3.

(a) multiply -2 to equation (1) and add it to equation (2). Equation (2) becomes

2y + 6z = 6

(b) Multiply 3 to equation (1) and add it to equation (3). Equation (3) becomes

-y + 3z = 9

And the original system is reduced to the system

x - 3y - 2z = 6

2y + 6z = 6

-y + 3z = 9

Now, we have to remove the second unknown, y, from new equation 3, using only the

new equation 2 and 3 (above).

a, Multiply equation (2) by 1/2 and add it to equation (3). The equation (3) becomes 6z =

12.

Therefore, our given system of three linear equation of 3 unknown is reduced to thetriangular system

x - 3y - 2z = 6


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2y + 6z = 6

6z = 12

Second Part

In the second part, we solve the equation byback substitutionand get

x = 1, y = -3, z = 2

In the first stage of the algorithm, the coefficient of x in the first equation is called the

pivot, and in the second stage of the algorithm, the coefficient of y in the second equationis the point. Clearly, the algorithm cannot work if either pivot is zero. In such a case one

must interchange equation so that a pivot is not zero. In fact, if one would like to code this

algorithm, then the greatest accuracy is attained when the pivot is as large in absolute value

as possible. For example, we would like to interchange equation 1 and equation 2 in theoriginal system in the above example before eliminating x from the second and third

equation.

That is, first step of the algorithm transfer system as

2x - 4y + 2z = 18

x - 4y + 2z = 18

-3x + 8y + 9z = -9

Determinants and systems of linear equations

Consider a system of n linear equations in n unknowns. That is, for the following system

a11x1+ a12x2+ . . . + a1nxn= b1 a21x1+ a22x2+ . . . + a2nxn= b2 . . . . . . . . . . . . . . . . . . . . . . . . .

an1x1+ an2x2+ . . . + annxn= bn

Let D denote the determinant of the matrix A +(aij) of coefficients; that is, let D =|A|. Also,

let Nidenote the determinants of the matrix obtained by replacing the ithcolumn of A by

the column of constants.


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Theorem.If D 0, the above system of linear equations has the

unique solution .

This theorem is widely known as Cramer's rule. It is important to note that Gaussianelimination is usually much more efficient for solving systems of linear equations than is

the use of determinants.
http://www.personal.kent.edu/~rmuhamma/Algorithms/algorithm.html

Linear Inequalities and Linear Equations

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