Solve Linear Algebra exam

April 9th, 2013 | Algebra | Comments Off

Introduction to solve linear algebra for exam:

The linear algebra is used to solve a process of the unknown quantity in which the equation is true.  The linear algebra here is denoted as a two variable function with 14 inches/foot. There are one or more variables available to represent a plane and to solve equation. A simple linear function have one independent variable ( y = a + bx ) which is used to trace a straight line in graph.

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Problem of linear algebra for exam:

A high school had 1300 students enrolled in 2004 and 1600 students in 2007. Conducting exam for students. If the student population P; when exam is conducted P grows as a linear function of time t, where t is the amount of years after 2004.
a) How many students will be enrolled in the school in 2009?
b) Solve a linear algebra function that relates to the student population to the time t.

Solution:

a) Consider the given information in pairs as (t , P). The year 2004 correspond to t = 0 and the year 2007 corresponds to t = 4, hence 2 ordered pairs

(0, 1300) and (4, 1600)

Since the population grows linearly with time t, we use two ordered pairs to find slope m for graph of P as follows

m = (1600 – 1300) / (7 – 4) = 100 students / year

If slope is m = 100 then the students population grows by 100 in every year. Solve the equation from 2004 to 2009 in 7 years then students population in 2009 will be

P(2009) = P(2004) + 8 * 100 = 1300 + 800 = 2100 students.

b) We know the slope and two points, use point slope form to find an equation for  population P as a function of t as follows

P – P 1 = m (t – t1)

P – 1300 = 100 (t – 0)

P = 100 t + 1300

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Practice problem on linear algebra for exam:

A high school exam had 1100 students enrolled in 2005 and 1800 students in 2006. Conducting exam for students. If the student population P, when exam is conducted P grows as a linear function of time t, where t is the amount of years after 2005.
a) How many students will be enrolled in 2010?
b) Solve a linear algebra function that relates to student population to the time t.

Why is it called a Right Angle

April 8th, 2013 | Geometry | Comments Off

Introduction to ‘why is it called a right angle’:

An angle is called right angle as the angle between them measures a 90.,or the 2 lines which makes that angle intersects perpendicularly.

A right angle is formed by two halves of a straight line. Si ,in a ray which have its endpoint on a line and the adjacent angles are equal,then that pair is of right angles. If we consider rotation,the right angle rotates,a quarter (that is 1/4 of a full circle)

The term is a calque of Latin angulus rectus; here rectus means “upright”, referring to the vertical perpendicular to a horizontal base line
Some important geometrical figures are perpendicular lines which are closely related, means the lines which form a right angle as they intersect,at the point of intersection,

These are mainly dealt with the matter of vectors. The basis of the branch trigonometry is the presence of the right angle in a triangle.some other angles are acute angles and obtuse angles.

why is it called a right angle-Euclid’s right angle

Having problem with Angle Sum of a Triangle keep reading my upcoming posts, i will try to help you.

we all are aware of the book Euclid’s ELEMENTS. The right angles are the fundamental thing in that book. He had defined acute angles and obtuse angles in terms of right angles.And if the sum of the two angles are 90 they are called complementary angles.

Euclid uses the rightangle to measure all other ,as all right angles are equal in measure.

As we go thorough the history,we can see all carpenters and masons know how to calculate the right angle .

It is by the technique of 3-4-5.the basic Pythagorean triplet.and they called it as rule of 3-4-5.

If one side is of measure 3 and other side measures 4 then the 3rd side (the hypotenuse)measures 5 units.This measurement can be done without any instruments,so quickly.

the law behind this technique is pythagoras theorem.

base2 +altitude2= hypotenuse2

in any right triangle.

that is the sum of the square of the base and square of the height in a right angled triangle is always equal to the square of the hypotenuse.

why is it called a right angle-different units for right angle

Math is widely used in day to day activities watch out for my forthcoming posts on What is Angle and Holt Geometry Answers. I am sure they will be helpful.

In the cartisian co-orinate system the x and y axis form a rigt angle at the point of intersection,that is at the origin.

the different units to measure the right angle are,

1.90° (degrees)
2.?/2 radians
3.100 grad (may called grade, gradian, or gon)
4.8 points (in a 32-point compass rose)
5.6 hours (astronomical hour angle)

Types of Polynomial

April 5th, 2013 | Math | Comments Off

Introduction to types of polynomials:

Types of the polynomials article deals with the definition of the polynomials and their three types.

Definition of polynomials:

Polynomial is an expression constructed with one or more variables and constant term using the arithmetic operations like addition, subtraction and multiplication. Depending up number of terms, the polynomials are classified in to three types.

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Types of polynomials and its degree

The polynomials are classified in to three types

  • monomials
  • Binomials
  • trinomials

Monomials:

The polynomial that contains only one term

For example:

  1. 5x
  2. 3xy

Binomials:

The polynomial that contains two terms

The two terms may be

  1. two variables terms
  2. Variable term and constant term

for example

  1. 2x+3y
  2. 2x+5

Trinomials:

The polynomial that contain three terms

The polynomial will contain two variable terms and one constant

For example:

1. 3x+4y2+2.

Model examples:

Problem 1:  3y2

Solution:

The given polynomial is monomial

3 is the coefficient

The degree of the monomial is 2

Problem 2: 3x2-4x

Solution:

The given polynomial is binomial

3 is the coefficient of x2

4 is the coefficient of x

The degree of the polynomial is 2

Problem 3: 4x2+5x+5

Solution:

The given polynomial is trinomial

4, 5, 5 are the coefficient of the polynomial

The degree of the trinomials is 2

Arithmetic operation in polynomials

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We can perform the addition and subtraction among the like terms.

 Like terms: the terms that have the same variables are said to be like terms.

3x and 4x are the like terms.

Addition:

Example 1: 3x+4x

Solution:

3x and 4x are the like terms

= 7x.

Example 2: 3x2+4x+2x2+6x

Solution:

Add the like terms

3x2+2x2+4x+6x

5x2+10x

Example 3: 4x2+4y+3+5y+7

Solution:

Add the like terms

4x2 +4y+5y+3+7

4x2+9y+10.

Subtraction:

Example 1: 3x – 4x

Solution:

3x and 4x are the like terms

= -x.

Example 2: 3x2-4x-2x2+6x

Solution:

Subtract the like terms.

3x2-2x2-4x+6x

x2+2x

Example 3: 4x2+4y+3-5y-7

Solution:

Subtract the like terms

4x2 +4y-5y+3-7

4x2-y-4.

Fraction help website

April 4th, 2013 | Number System | Comments Off

Introduction to fraction help website:

Fraction:

A fraction is a number that can represent part of a whole.  The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on.

Website:            

A website is a collection of related web pages, images, videos or other digital assets that are addressed relative to a URL.

(Source : Wikipedia)

Our tutor vista website is the best website that provides help for all the subjects and all grade students. In this article we are going to see about help on fractions.

Help for fractions:

In this section we are going to see types of fractions.

In general, there are three types of fractions.

1. Proper fraction

2. Improper fraction

3. Mixed fraction

Proper fraction help:

The fraction is of the form a/b, where a and b are integers and b is always greater than a is called as proper fraction.

Example:

2/3, 56 /78, 100 /101 ……

Improper fraction:

The fraction is of the form a/b, where a and b are integers and a is always greater than b is called as improper fraction.

Example:

3/2, 78/56, 101/100 …….

Mixed fraction:

The fraction is of the form c a/b, where a, b and c are integers. Here b is divisor, a is remainder and c is called as quotient.

Example:

3 4/5, 6 19/34 …..

Basic operations on fraction:

All the basic arithmetic operations can be applied for fractions. In this section we are going to see addition and multiplication help for fraction.

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1. Addition of fractions:

To add a fraction we need the common denominator.  We can add the fractions with same denominator.

Example 1:

Add   3/4 + 6/4

Solution:

3/4 + 6/4 = (3+6) /4

= 9/4

Example 2:

Add 5/7 + 9/6

Solution:

To add a fraction we need to make common denominator,

5/7 + 9/6 = (5 * 6) / (7 * 6) + ( 9 *7) / ( 6 * 7)

= 30/42 + 49 /42

= (30+49) / 42

= 79 / 42

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2. Multiplication of fractions:

Just like integers we can multiply the fractions.

Example:

Multiply 3/8 * 24 / 27

Solution:

Given, 3/8 * 24 / 27

3/8 * 24 / 27 = (3 * 24) / ( 8 *27)

= 72 / 216

Divide by 72 on both numerator and denominator,

= 1/3

Logarithmic properties

April 3rd, 2013 | Math | Comments Off

In this page we are going to learn about logarithmic properties .In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce that number.

For example, the logarithm of 10000 to base 10 is 4, because 4 is the power to which ten must be raised to produce 10000: 104 = 10000, so log1010000  = 4. Only positive real number has real number logarithms; negative and complex numbers have complex logarithms.                                                                                                                                           (Source.wikipedia)

Below are the properties of logarithmic given follow -

  • Product rule: If a, m and n are positive numbers and a ?1, then

loga(mn) = logam +logan


  • Quotient rule: If m, n and a are positive numbers and a ? 1, then,
log a(`m/n` ) = log am –loga n
  • Power rule: If a and m are positive numbers, a ? 1 and n is a real number, then

logamn =nlogam

  • Change of base rule: If m, n and p are positive numbers and n ? 1, p ? 1, then

lognm = logpm / logpn

  • Reciprocal rule: If m and n are the positive numbers other than 1, then

logmn =1 / lognm

Examples of natural logarithms

Below  are the examples of  logarithmic properties -

Example 1:

Reduce:log3 27 + log3 729 (ii) log5 8 +1 /  log5 `1/1000`

Solution:

Since the expression is the sum of the two logarithms and the bases are equal, we can apply the product rule

(i) log3 27 +log3 729 = log 3 (27*729)

= log3 (33*36)

= log3 39 = 9 log33 = 9*1 =9

(ii) log58+log5(1/1000)  = log5 (8*1/1000)

= log 5`(1/125)`

=log 5(1/  53)

= log5(5-3)

= -3 log55 = -3*1= -3

Example 2:

Reduce: log7 98-log714

log7 98-log714 =log 7(98/14)

= log77 =1

Example 3:

Solve .log62x – log6(x+1)

Solution: Using the quotient laws, we can write the equation as log6 (2x / x+1) Changing into exponential form, we get

(2x / x+1) = 60

2x = x + 1 or x = 1.

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Example 4:

Solve .log10 (2x+50) = 3

Solution: Writing the equation in the exponential forms, we get

2x + 50 = 103 = 1000 or 2x = 1000 ? 50 = 950 or x = 475.

Example 5:

Solve .log3 (7-x) – log3 (1-x) =1

Solution: Using the quotient laws, the equation can be written as log3(7-x / 1-x) = 1

Writing in the exponential form, we get

7-x / 1-x = 31=3

7-x= 3(1-x)

7-x = 3-3x

2x = ?4

x = ?2.

Example 6:

Solve:

log2(log3 x) = 2

Solution:

Put y =log3x Then the equation becomes log2y =2. Writing the equation in the exponential form, we get or y=22 =4.log3x =4 Again, writing in the exponential form, we get x =34

X =81

Triangle Congruence Problems

March 28th, 2013 | Geometry | Comments Off

CONGRUENT TRIANGLES : Two triangles are congruent if and only if one of them can be made to superpose on the other so as to cover exactly.

some elementary theorem of congruence which are often used

1) SIDE-ANGLE-SIDE

2)ANGLE-SIDE-ANGLE

3)ANGLE-ANGLE-SIDE

4)SIDE-SIDE-SIDE

5) RIGHTANGLE-HYPOTENUSE-SIDE

Understanding Congruent Triangles is always challenging for me but thanks to all math help websites to help me out.

1) P and Q are points on equal sides AB and AC of an isosceles triangle ABC such that AP=AQ prove that PC = QB

2)In ? ABC and ? PQR , AB =PQ  BC = QR ; CB and RQ  are  extended to X and Y  respectively .`-<` ABX = `-<` PQY PROVE THAT ?ABC `~=` ? PQR

3)AB  is a line segment .AX  and BY are two equal line segments drawn opposite sides of line  AB  such that  AX is parallel to BY if line segments AB and XY itersect each other at point P .prove that  ? APX `~=` ? BPY  .  and line segments AB and XY bisect each other at  P.

PROBLEM -4

Hari wishes to determine the distance between two objects A and B ,  but  there is an obstacle between these two objects which prevent him from making a direct measurement. He devises an ingenious way to overcome this difficulty.Firstbhe fixes a pole at a convenient point O so that from  O , both A and B are visible.then he fixes another pole at the point D on the line AO (produced) such that AO =DO in a similar way he fixes a third pole at the point  C .on the line BO.(produced ) such that BO = CO. Then he measures CD which is equal to 170 cm. prove that the distance between the objects A and B is also 170 cm.

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PROBLEM -5

? ABC  IS isosceles with AB = AC  prove that altitudes BD and  CE  of triangle are equal ?

problem -6

?  ABC is  an isosceles triangle with AB = AC , BD and CE are two medians of triangle , prove that BD = CE .

solving area of a rectangle

March 1st, 2013 | Geometry | Comments Off

Introduction to solving area of a rectangle:

A quadrilateral is a called rectangle, if it is a parallelogram and each angles measures 90 degrees.

or explaining in simpler terms:

  • A rectangle is a four sided and closed figure
  • In a rectangle all opposite sides are parallel and equal
  • and also in a rectangle all the adjacent sides are perpendicular.

A rectangle is a quadrilateral and also a parallelogram.

Quadrilateral is a four sided closed figure.

Parallelogram is a quadrilateral with opposite sides parallel and equal.

If all the sides are equal the rectangle becomes a square.

Properties of a rectangle:

1.Opposite sides are equal.

2.All angles are right angles (90 degrees).

3.Diagonals of a rectangle are equal and bisect each other. ( Or both diagonals intersect each other at their mid points.)

Solving area of a rectangle

Area of a rectangle is its length multiplied by width.(Length x Width)

Let us consider a rectangle with length x units and width y units.

Rectangle

Area of the above rectangle = x*y square units.

Example 1:The area of a rectangle with length 5 feet and width 4 feet = 5 * 4 = 20 sq. feet.

Example 2:

What is the area of a rectangle with length 3m and width 3cm?

Ans: first of all we have to make the unit of both length and width same.

here lets convert the length (3m) to centimeters:

so length = 3m =3 x 100 cm = 300cm

width = 3cm

so area = 300 x 3 = 900 square centimetres.

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Example on solving area of rectangle:

1) A white paper has a size of length 20 cm and width 15 cm. Steve has drawn a rectangle in the center of the paper with length 10 cm and width 5 cm and painted the entire rectangle in yellow color.Find the area of the yellow colored portion and area of white portion.

Answer:

we know that area of a rectangle is length x width.

here the given paper has the shape of a rectangle.

so its area = 20cm x 15cm = 300cm² .

area of the yellow colored rectangle = 10cm x 5cm = 50cm².

so the area of the white portion = 300cm² – 50cm² = 250cm².