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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