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

Example of inequalities are:

i.
x1

ii. x + y + 2z > 16

iii. p^{2}
+ q^{2}
1/2

iv. a^{2} + ab > 1

ii. x + y + 2z > 16

iii. p

iv. a

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

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

2. If ab and c is positive, then acbc.

For example, 23 implies 2(4) 3(4).

3. If ab and c is negative, then acbc.

For example, 39 implies 3(-2)9(-2).

4. If ab and bc, then a c.

For example, -1/22 and 28/3 imply -1/28/3.

For example, 1 is a solution of 2x + 37 since 2(1) + 3 = 5 and 5 is less than and equal to 7.

By a solution of the two variable inequality x - y5 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 - y5 because 2-1 = 1 and 15.

By a solution of the three variable inequality 2x - y + z3 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 + z3.

A solution of an inequality is said to satisfy the inequality. For example, (2, 1) is satisfy x - y5.

Two or more inequalities, each with the same variables, considered as a unit, are said to form a system of inequalities. For example,

x0

y0

2x + y4

y0

2x + y4

Note
that the notion of a system of inequalities is analogous to
that of a solution of a system 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.

An inequality in two variable x and y
describes a region in
the x-y plane (called its graph), namely, the set of all points whose
coordinates
satisfy the inequality.

The y-axis
divide, 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-axis divides the xy-plane
into two half-planes.

- Upper
half-plane

In which inequality y > 0 is true. - Lower
half-plane

In which inequality y < 0 is 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.

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

ax
+ by = c

Solution of Linear Equation

A solution of the equation consists of a pair of number, u = (k

Geometrically, any solution u = (k

a_{1}x
+ b_{1}x
= c_{1}

a_{2}x + b_{2}x = c_{2}

a

Geometrically, there are three cases of a simultaneous solution

- If the system has exactly one solution, the graph of the linear equations intersect in one point.
- If the system has no solutions, the graphs of the linear equations are parallel.
- 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 linear equations are proportional.

OR
=> a_{1}b_{2}
-_{ }a_{2}b_{1}
= 0 => = 0

The system has no solution when

The solution to system

a_{1}x
+ b_{1}x
= c_{1}

a_{2}x + b_{2}x = c_{2}

a

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

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 be solved 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)

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

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

6x + 4y = 16

-6x + 15y = 3

-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 solution to the system.

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

a_{11}x_{1}
+ a_{12}x_{2}
+ . . .
+ a_{1n}x_{n} = b_{1}

a_{21}x_{1} + a_{22}x_{2}
+
. . . + a_{2n}x_{n} = b_{2}

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

a_{n1}x_{1} + a_{n2}x_{2}
+ . . .
+ a_{nn}x_{n} = b_{n}

a

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

a

x_{1}
= k_{1},
x_{2}
= k_{2}, . . . , x_{n}
= k_{n}

or equivalently, a list of n numbers

u = (k_{1}, k_{2},
. .
. , k_{n})

is called a solution of the system if, with k

The above system is equivalent to the matrix equation.

or, simply we can write A × = B, where A = (a

The matrix is called the coefficient matrix of the system of n linear equations in the system of n unknown.

The matrix is called the augmented matrix of 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 b

If a

a

a

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

a

a

Where |A| = a

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, x

x_{n}
= b_{n}/a_{nn}

2. Second, we substitute the value of x

.

3. Third, we substitute these values for x

.

In general, we determine x

.

Gaussian elimination is a method used for finding the solution of a system of linear equations. This method consider of two parts.

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

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

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 the triangular system

x - 3y - 2z = 6

2y + 6z = 6

6z = 12

2y + 6z = 6

6z = 12

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 equation is 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 the original 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

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

a_{11}x_{1}
+ a_{12}x_{2}
+ . . .
+ a_{1n}x_{n} = b_{1}

a_{21}x_{1}
+ a_{22}x_{2}
+
. . . + a_{2n}x_{n} = b_{2}

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

a_{n1}x_{1}
+ a_{n2}x_{2} + . . .
+ a_{nn}x_{n} = b_{n}

a

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

a

Let D denote the determinant of the matrix A +(a

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 Gaussian elimination is usually much more efficient for solving systems of linear equations than is the use of determinants.