Definition of Slope in Math

Introduction for definition of slope in math:

Definition:

The slope of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line. Given two points (x1,y1) and (x2,y2) on a line, the slope m of the line is

Slope (m) = `(y_2 - y_1)/(x_2 - x_1)`

Source: Wikipedia

The definition of slope in math example problems and practice problems are given below.

Example Problems for Definition of Slope in Math:

Example problem 1:

Determine the slope of the line, 5x - y = -13

Solution:

The slope intercept form of a line is y = mx + b, where m is the slope of the line.

Given equation is in the form of ax + bx + c = 0. To find the slope of the line, we have to convert the equation from general form to slope intercept form.

The given equation can be converted from general form to slope intercept form as follows,

5x - y = -13

Subtract 5x on both sides,

5x - y - 5x = -13 - 5x

-y = -13 - 5x

Divide by (-1) on both sides.

`-y/(-1) ` = -`(13)/(-1)` - `(5x)/(-1)`

y = 13 + 5x

y = 5x + 13

Now the equation is in the form of y = mx + b, so the slope of the given line is 5

Example problem 2:

Determine the slope of a line, which contains the points A (4, -3), B (-2, 8).

Solution:

The slope of a line which contains two points (x1, y1) and (x2, y2) is given by,

Here, x1 = 4, x2 = -2, y1 = -3, y2 = 8.

Slope of the line, m = ` (y_2 - y_1)/(x_2 - x_1)`

m = `(8 + 3)/( -2 - 4)`

m = `-11/6`

Slope of the line (m) = `-11/6`

So, the slope of a line, which contains the points A (4, -3), B (-2, 8) is `-11/6`

Practice Problems for Definition of Slope in Math:

Practice problem 1:

Determine the slope of the line, 8x - y = -60

Answer: Slope (m) = 8

Practice problem 2:

Determine the slope of the given points: (12, -4) and (2, 8)

Answer: Slope (m) =  `-6/5`

Corollary of Theorem

Corollary of theorem - Introduction:

In math a corollary normally follows a theorem. The exercise of the expression corollary, slightly than proposal or theorem, is basically subjective. Proposal B is corollaries of proposal A if B can readily be assume from A, but the meaning of gladly varies depending on the author and context. The importance of the corollary is frequently measured secondary to that of the primary theorem; B is suspect to be termed a corollary if its math consequences are as important as those of A. Occasionally a corollary has a proof that explicate the derivation; on occasion the derivation is considered to be self-evident.

Corollary of some Theorems:

Corollary of Theorem

Let Y ~ B (n, `pi` ), where `pi` is called the population proportion, n is the sample size and Y is the number of “success”s in the sample. Then the sample proportion is p = `Y/n` . Given these,

`lim_(x->oo)P(|y/n - pi| <= epsi)= lim_(x->oo)P(|p-pi|<=epsi) =1`

This states that as the sample size increases the sample proportion converges in probability to the population proportion, that is, p->`pi` .

Theorem

Strong Law of Large Numbers or almost sure convergence

Let X1, X2,.. Xn be a random sample from a population (of X’s) with mean E(X) = mX and a finite variance, Var(X) = `sigma^2_x < oo` . Let `bar(X) = 1/n sum_(i=1)^n X_i` . Then, for any positive real number e,

`P(lim_(n->oo) |bar(X)- mu_x| <= epsi =1 or p(lim_(n->oo)) | bar(X) - mu_x|>=epsi) = 0`

Angle Sum Theorem:

The sum of the angle measures in any triangle is 180°

PROVE IT!

x + x + y = 180° (Triangle)

y + y + y = 180° (Line)

y = x (Alternate Interior Angles)

Corollary to the Angle Sum Theorem:

If ∆ABC is any triangle, then an exterior angle of ABC has the same measure as the sum of the measures of the two nonadjacent interior angles.

PROVE IT!

m `angle` 1 + m `angle` 2 = m `angle` 3

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Altitude Similarity Theorem

The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.

`Delta` ABC ~ `Delta` ACD ~ `Delta ` CBD

Corollary 1

The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.

`(AD)/(CD) = (CD)/(DB) CD = sqrt(AD(DB)) `

Corollary 2

The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse.

`(AD)/(AC) = (AC)/(AB') , (BD)/(CB) = (CB)/(AB)`

Present Value Equation

Introduction for present value equation:

Present value is the given value to find the future value for the payment. Finding present value is used for the flow of the cash in different time. The formula for finding the present value is given below.

Present value = `"Future value"/(1+i)^t`

Where “i” is rate of annual interest and “t” is number of years. I like to share this Solve Absolute Value Equations Calculator with you all through my article.

Examples for Present Value Equation:

Example 1 for present value equation:

Martin needs to deposit $40000 in his bank account to buy a new car in two years. So he want to deposit his money in two years with the rate of interest is 6%. Calculate his present value.

Solution:

Future value is $10000.

Number of years (t) is 2.

Rate of annual interest (i) is 6% of `6/100` = 0.06.

Present value = `"Future value"/(1+i)^t`

Present value = `40000/(1+0.06)^2`

Present value = `40000/(1.06)^2`

Present value = `40000/1.1236`

Present value = 35599.86

Therefore, Martin’s Present value is $ 35,599.86

Example 2 for present value equation:

Paul needs to deposit $45000 in his bank account to buy a new Laptop in two years. So he want to deposit his money in three years with the rate of interest is 8%. Calculate his present value.

Solution:

Future value is $45000.

Number of years (t) is 3.

Rate of annual interest (i) is 8% of `8/100` = 0.08.

Present value = `"Future value"/(1+i)^t`

Present value = `45000/(1+0.08)^3`

Present value = `45000/(1.08)^3`

Present value = `45000/1.259712`

Present value = 35722.45

Therefore, Paul’s Present value is $35,722.45

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Practice Problems for Present Value Equation:

Problem 1 for present value equation:

John needs to deposit $50000 in his bank account to buy a new house in two years. So he want to deposit his money in four years with the rate of interest is 7%. Calculate his present value.

Solution:

John’s present value is $ 38,144.76

Problem 2 for present value equation:

Eric needs to deposit $55000 in his bank account to buy a new computer in two years. So he want to deposit his money in five years with the rate of interest is 9%. Calculate his present value.

Solution:

Eric’s Present value is  $ 35,746.23

Problem 3 for present value equation:

Belix needs to deposit $60000 in his bank account to buy a new car in two years. So he want to deposit his money in six years with the rate of interest is 10%. Calculate his present value.

Solution:

Belix’s present value is $ 33,868.44

Pi Fraction Approximation

Introduction for pi (π) fraction approximation:

In this article we shall discuss about pi fraction approximation. A number Π (sometimes note down as pies) is a numerical constant whose value is the ratio of every circle's circumference to its diameter in Euclidean space; this is the like worth as the ratio of a circle's area to the four-sided figure of its radius. It is just about equivalent to 3.141593 in the common decimal notation.

I Need Help with Pi (π) is an Irrational Number:

The need help with pi (π) is an Irrational number. So, Irrational number is given below that,

Designed for a lot of centuries previous to the real evidence, mathematicians have consideration that pi was an irrational number. The initial effort at confirmation was through Johann Heinrich Lambert in 1761. From side to side a complex technique he recognized that if x is rational, `tan(x)` have to be irrational. It goes after that if `tan(x)` is rational, x have to be irrational. Because `tan (pi/2) = 1` , `pi/2` should be irrational; thus, pi data should be irrational.

A lot of people saying Lambert's evidence as too simplified an answer for such a complex and long-lived problem. In 1794, though, A. M. Legendre establishes one more evidence which reverse Lambert up. This original evidence as well go as far as to establish that π2 (pi2) were also irrational.

I Need Help with Notation for Pi (π) Fraction Approximation:

The need help with notation for pi (π) fraction approximation is given below that,

We cannot note down an easy division that equals Pi. The pi fraction approximation denoted as `22 / 7.`

At the present, infinite series value of pi (π) fraction approximation is given below that π (Pi) = `22 / 7 ` = 3.14159265358979323.

The accepted rough calculation for numbers like pi (π) data = 3.141592653589793238462643383 is closed but not precise. Please express your views of this topic derivative of cosine by commenting on blog.

Example for numbers like pi (π) data irrational number π (Pi) = 3.14159265358979323846264…,

The pi (π) fraction approximation value is `22/7.`

Patterns and Functions Math

Introduction to patterns and functions math:

The various number patterns in mathematics are formed by the functions that define the relation between the consecutive numbers in the series. If the interval between the numbers remains same through the series then the series is called as recursive series. In the following article we will see in detail about the topic recursion and special sequences online study.

More about Patterns and Functions Math:

Recursion:

The recursive series is not just a single series but there are various series that can be formed by the repetition of the condition that defines the numbers in the series. In the recursive series they may be formed by the repeating interval or the repeating conditions that defines the consecutive numbers of the series. All the recursive series requires two important parameters, the first value of the series and the rule or the condition that defines the occurrence of the numbers in the series.

Fibonacci numbers:

The Fibonacci series starts with two numbers and the rest of the numbers in the series are formed by the rule that the numbers in the series are the sum of the previous two Fibonacci numbers. The function used to form the numbers in the Fibonacci series is given by,

`F_n = F_(n-1) + F_(n-2)`

Polygonal numbers:

The polygonal numbers are the numbers which are represented as dots and arranged in the form of a particular polygon. And all the polygons have such kind of number sequences. All the polygonal numbers starts with 1 and the next numbers in the series is calculated by increasing one dot in consecutive arms of the polygonal shape and filling the rest of the arms.

Example Problem on Patterns and Functions Math:

1. Find the first five numbers in the recursive series with the first number `a_0 = 10` and `a_(n+1) = a_n +3` .

Solution:

`a_0` = 10

`a_1` = `a_0` + 3 = 10+3 = 13

`a_2` = `a_1` + 3 = 13+3 = 16

`a_3` = `a_2` + 3 = 16+3 = 19

`a_4` = `a_3` + 3 = 19+3 = 22

Practice problem on patterns and functions math:

1. Find the first 6 numbers in the recursive series with `a_0 = 20` and `a_(n+1) = (a_n)-(2*n)+3` .

Answer: 20, 21, 20, 17, 12, 5.

Empirical Formula Statistics

Introduction to Empirical formula Statistics:

The science of making effective use of numerical data values related to groups of individuals or experiments is called Statistics. Empirical Statistics denotes the information gained by means of observation, experience, or experiment. Empirical Statistics fully depends on the evidence or consequences that are observable by the senses.It refers to the use of working hypothesis that are tested using observation or experiment.

Formula for Finding Empirical Statistics:

The Formula to find the Empirical Statistics is listed as below

Formula for Mean (or) Expected value:

μ = E[x] = np

Formula for Standard deviation:

σ =`sqrt(npq)`

Formula for Variance:

E[x^2] = σ^2 = npq

Example Problems for Solving Empirical Formula Statistics:

Example 1:

A coin is flipped for 68 times. Determine the mean, variance and standard deviation with the help of Binomial distributions.

Solution:

Let coin flipped for 68 times. So n =68.

If we flip a coin means, possibility of getting head is p =`(1)/(2)`

Probability of achieving tails is represented by q.

q = 1 – p               [p + q = 1]

q = 1 –`(1)/(2)`

q =.`(1)/(2)`

Mean:

μ = E[x] = np

=` 68(1/2)`

μ = E[x] =34

Variance:

E[x2] = σ^2 = npq

=`68(1/2)(1/2)`

E[x2] = σ^2 = 17

Standard deviation:

σ =`sqrt(npq)`

σ = `sqrt(68*(1/2)*(1/2))`

= `sqrt(17)`

σ = 4.12

Example2:

A coin is tossed for 48 times. Find the mean of tails, variance and the standard deviation with the help of Binomial distribution. Please express your views of this topic horizontal asymptotes by commenting on blog.

Solution:

Let coin tossed for 48 times. So n = 48.

If we toss a coin means, probability of getting tail is p = `(1)/(2)`

Probability of getting head is represented by q.

q = 1 – p                     [p + q = 1]

q = 1 –`(1)/(2)`

q = `(1)/(2)`

Mean:

μ = E[x] = np

= `48(1/2)`

μ = E[x] = 24

Variance:

E[x2] = σ^2 = npq

= `48(1/2)(1/2)`

= 48

E[x2] = σ^2 =12

Standard deviation:

σ =`sqrt(npq)`

σ = `sqrt(48*(1/2)*(1/2))`

=` sqrt(12)`

σ = 3.46

Statistics Sample Size Calculator

Introduction to statistics sample size calculator:

Let us see about statistics sample size calculator. The calculator is one of the machines which are helped to calculate the statistics sample size. The statistics sample size is the parts of the population that supports us to illustrate the inferences about the population. Gather examine of the collection information about the population is not probable and it is time unbearable and contented. Thus, we involve an appropriate sample size so that we can generate inferences about the population foundation on that sample size.

Formula for the Staitstics Sample Size Calculator:

Let us see the formula for the statistics sample size.

Sample size = `(z^2 * (p) * (1-p)) / c^2` .

Description:

Z defines confidence level.

P defines Percentage picking a choice.

C defines confidence interval.

This is used to calculate the sample sixe by using the calculator.

Examples of sample size calculator:

Let us see some examples of statistics sample size calculator.

Example 1:

Determine the statistics sample size by using calculator where the confidence level is 95% and interval is 0.4 and population is 24%.

Solution:

The formula for the sample size = z^2 * (p) * (1-p) / c^2.

Step 1: Enter the confidence level in the calculator. The confidence level must be either 95% or 99%. So, choose the appropriate the confidence level.

Step 2: Enter the confidence level.

Step 3: Enter the population value.

Step 4: Enter the calculate button, you will get the statistics sample size by using sample size formula.

Step 5: Press the clear option, put different values and find the sample size.

These are the steps used to calculate the sample size by using calculator. Please express your views of this topic Simplifying Fractions Calculator by commenting on blog.

One more Example of Statistics Sample Size Calculator:

Determine the statistics sample size by using calculator where the confidence level is 95% and interval is 0.4 and population is 27%.

Solution:

The formula for the sample size = z^2 * (p) * (1-p) / c^2.

Step 1:

Enter the confidence level in the calculator. The confidence level must be either 95% or 99%. So, choose the appropriate the confidence level.

Step 2: Enter the confidence level.

Step 3: Enter the population value.

Step 4: Enter the calculate button, you will get the statistics sample size by using sample size formula.

Step 5: Press the clear option, put different values and find the sample size.

These are the steps used to calculate the sample size by using calculator.

Dividing decimals

Decimal numbers is one of the types in our number system. When a number is not a whole number, we express the number as a sum of whole number part and the fraction part. In the fractional form we use the fractions for the fraction part. But for easier working, the fraction is converted to the parts of powers of 10 and such parts are described along with the number with a dot ‘·’ separating them. This dot is called as the decimal points.

Like in all other cases, decimals are also subject to all basic algebraic operations. One of the operations is dividing decimals by whole numbers or dividing decimals by decimals. The rules for dividing decimals are very simple to follow. Let us start first discussing how to divide decimals by whole numbers.

The division of decimals by whole numbers is almost same as we divide whole numbers by whole numbers. But do not forget to place the decimal point in the quotient as soon as you finish the division with whole number part. That is, at this point place the decimal point and carry forward the division as before. For example, while dividing 16.535 by 5, the division of whole number part is complete when 16 is divided by 5. The quotient at this stage is 3 and the remainder 1 is carried forward to group with the digit 5 after the decimal point. Therefore the quotient 3 now should be followed by the decimal point and only after that you can enter the next quotient

Now let us concentrate on how to divide decimals by decimals. The simplest dividing decimals practice here is convert the decimal numbers of both the dividend and divisor to whole numbers by suitable powers of 10. If required use 0s as space holders. But due care must be taken to place the decimal point after dividing the last digit/digits and if the division is not complete. For example, (7.236)/(4.02) can be tried as (7236)/(4020) = 1.8

We need to face dividing decimal numbers in our daily life. Most of the prices are not whole numbers and are expressed as decimal number of dollars in our currency. To counter such situations, one can practice a few examples of dividing decimals word problems. For example, the price of a pack of 5 pairs of socks is labeled as $4.5. If you want to know what is the cost of one pair of socks, you must be familiar with the division that (4.5)/(5) = (45/5) = (90/10) = 0.9. Thus one costs $0.90.

Solve the Equation in the Real Number System

Introduction of solve the equation in the real number system:

Positive or negative, small or large, whole or decimal numbers are called real numbers.  Real number systems are represented by R.  Rational and irrational numbers are also called real numbers. Rational numbers are referred to as an integer or fraction. An irrational numbers cannot be expressed as a rational numbers. Example for irrational numbers are pi(3.14) and sqrt2. A real number system is a set of numbers that contains more operations such as addition, subtraction,  multiplication and so on. I like to share this Partial Differential Equation with you all through my article.

Example Problems – for Solve the Equation in the Real Number System

Example problem 1 - for solve the equation in the real number system

Solve the equation and find the value of x and y.

2x + y = 2

3x + y = 4

Solution:

2x + y = 2     -----------   1

3x + y = 4     -----------    2

2x + y = 2

3x + y = 4

_________

-x = -2

x = 2

substitute x = 2 in 1st equation

2(2) + y = 2

4 + y = 2

y  = -2

Answer: x=2, y= -2.

Example problem2 - for solve the equation in real number system

`sqrt(2)` is an irrational number. consider the equation of` x^2` +6x+1=0

` x^2` +6x+1=0

Solution:

-b`+-(sqrt(b^2 - 4ac))/(2a)`

-6 `+-` `(sqrt(6^2-4(1)(1)))/(2(1))`

-6 `+-` `(sqrt(36-4))/(2)`

-6 `+-` `(sqrt(32))/(2)`

-6 `+-` `(4sqrt(2))/(2)`

-6 /2 `+-` `(4sqrt(2))/(2)`

-3 `+-` `2sqrt(2)`

-3+`2sqrt(2)` , -3-`2sqrt(2)`

Both solutions are irrational numbers.

Example Problem3– for Solve the Equation in the Real Number System

Solve the equation and find the value of x, y and z.

2x + 4y + 4z = 0.

x + y + z = 6

x + 2y + 3z = 2

Solution:

2x + 4y + 4z = 0    -------------   1

x + y + z = 6          -------------    2

x + 2y + 3z = 2      -------------    3

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Take 2nd and 3rd equation and solve the problem

x + y +  z  = 6

x + 2y + 3z = 2

_______________

-y -2z = 4   -------------   4

_______________

Take 1st and 2nd equation

x + y + z =  6    ----------  Multiply 2 in 2nd equation

x = 12 Then, we get 2x + 2y + 2z  = 12.

2x + 4y + 4z = 0

2x + 2y + 2z = 12

________________

2Y + 2z = -12    ------------  5

________________

-y -2z = 4

2y + 2z = -12

_____________

Y = -8

_____________

Substitute y= -8 in 5th equation.

2(-8) + 2z = -12

-16 + 2z = -12

2z = -12 + 16

2z = 4

Z = 2

Substitute y=-8 and z=2 in 2nd equation

x + y +z = 6

x + (-8) +2 = 6

x -8 + 2 = 6

x – 6 = 6

x = 12

Practicing Problem - for Solve the Equation in the Real Number System

Practicing problem – for solve the equation in the real number system

Solve the equation and find the value of a, b, c.

2a + b = 8

2a + 2b =  4

Answer:  a = 6 , b = -4.