Algebraic Equations

Algebraic Equations Definition is given as equations that are a sum of number of terms where each term is the multiplication of some constant term and positive integral power of some variables. Algebraic equations can be divided into following categories as:
Linear equations: these include all equations with degree 1 which means the maximum power of variables can be 1 only. For example: x+5=12.

Quadratic equations: these are the equations of degree 2. Graphically these are represented by a U shaped parabola. Mathematically they are represented as: ax2 + bx + c=0 where a  0.
Exponential equations: these are equations where a term includes variable as the power of constant value for example: ex + e =0. Or 2x + x =2 etc.

Logarithmic equations: these equations involves a logarithmic term which is given as: loga x. Here ‘a’ cannot be 1 and a >0, x>0.  Let us see some methods of How to Solve Algebra Equations through some Algebraic Equations Examples now.

1) Linear equations: these equations are very easy to solve. Solution of the equation means to find out the value of variable that satisfies the given equation. Linear equations can be solved by isolating the variable of the equation. For ex:
2x+ 6 = 5.
to isolate x we can move 6 to RHS to get: 2x=5-6 = -1.
now a can be moved to RHS as: x= -1/2.

2) Quadratic equations: there are many methods to solve these equations like factoring, completing the square, quadratic formula and graphing. Here we will be quadratic formula to solve a quadratic equation.
The formula to solve a quadratic equation ax2 + bx + c=0 is given as:
x =
Example: solve 2x2 – 5x +2 =0.
By using above formula we have: a= 2, b =-5 and c =2.
x =

x =
x =
x=
So we get two solutions here as: x=(5+3)/4 and (5-3)/4 or 2, ½.

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3) Exponential equations: to solve these we first isolate exponential term and then take logarithm  at both the sides of equation.
Example: e2x - 3ex+2=0.
This can be written as: (ex-1)(ex-2)=0
Equate both terms of product by zero:
ex - 1=0
ex =1
=> taking logarithm on both sides we get: ln(ex) = ln(1)
=> x (ln(e)) = 0
=>x.1=0. Hence x=0.
ex - 2=0
Similarly solving this we get x=ln(2)=0.69314 approx.

4) Logarithmic Algebra Equation is solved by taking exponential.
Example: ln(x)=3
eln(x) = e3
x=e3.
x=20.085537 approx.

Grade 10th Math

Introduction to grade 10th math:

Mathematics is an essential tool used in throughout the entire world in many areas, like science, medicine, engineering, and social sciences. The 10th grade mathematics covers number system, algebra 1, algebra 11, pre calculus, calculus, statistics, geometry, trigonometry, and advanced trigonometry. In this article we shall discuss about 10th grade math problems and its solutions.

Grade 10th math example problem:


Example 1:

Solving the coefficient of the equation

`(1-(2)/((x-4))) = 8`

Solution:

This equation in the form of rational expression, multiplying the expression by the denominator value (x -4)
(x - 4) (1 - 2 / (x - 4)) = (x - 4)8

Simplifying the equation
(x - 4) - 2 = 8 (x - 4)

Multiplying the factors and arrange the like terms.
x - 6 = 8x - 32

Adding the value -8x to both sides and simplifying the equation

-7x - 6 = - 32

Adding the value + 6 to both sides
-5x = - 25

Solving for x

x = 25 /5.

x=5.

Example 2: Solve the inequality -4 < 2(x+4)-2<18 nbsp="" p="">
Solution:

Given inequalities

-4 < 2(x+4)-2<18 nbsp="" p="">
Multiplying the factor values and arranges the like term

-4<2x nbsp="" p="">-4<2x nbsp="" p="">Subtracting the value 6 on both sides of equation

-4-6<2x nbsp="" p="">-10<2x nbsp="" p="">
Dividing all terms by 2

-5
Conclusion:

The solution consists of all real number the interval (-5, 6)

Example:

A hemispherical bowl of radius 20 cm is contains with soap cream paste. This ice cream paste is made into cylindrical cakes each of radius 2 cm and height 2 cm, how many cakes do we get?

Solution:

Radius of the hemisphere = 20 cm;

Volume of the hemisphere =`3/ 2` `pi` r3 cubic units;

Therefore Volume of the soap paste =`3/2` x `pi ` x 20 x 20 x 20 cm3.

Radius of the cake = 2 cm;

Height of the soap cake = 2 cm;

Volume of the cake =pi x 2 x 2 x 2cm3.

Number of soap cakes = Volume of the soap cream paste / Volume of the cake

=   `(2/3 pi * 20*20*20)/(pi*2*2*2)`

Therefore we get 667 cakes out of the given paste.

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Grade 10th math practice problem:


Problem 1:

Solving the coefficient of the equation `(1-2/ (x-3)) ` =6

Answer:

x=13/5

Problem 2:

Solve the inequality -2< 2(x+6)-3<17 nbsp="" p="">Answer:

The solution consists of all real number the interval (-6, 4)

Learn Odds in Math

Introduction to learn odds in math:

A number which is not divisible by two is called as odd number in math. This is considered as whole number. The odd numbers are represented in set as { 1, 3, 5, 7, 9,……} and it does not contain multiples of two in math. We can learn the odds by identify the last digit that is if the last digit is 1,3 ,5 ,7 and so on means that number is odd.


Explanation for odd in math


Learn odd in math:

We can learn odd number’s arithmetic operations by using some results. They are,

In addition and subtraction:

If we add the even number and odd number means the resultant value is odd.For example add the numbers 8 + 1 = 9.

If we add the two odd numbers means the resultant value is even. For example add the 5 + 7 = 12.

In multiplication:

If we multiply the even and odd numbers means the resultant value is even. For example multiply the 6 x 7 = 42.

If we multiply the two odd numbers means the resultant value is odd. For example multiply the numbers 3 and 5 and we get the answer as odd value 15.

In other words:

The odd numbers are also called as gnomonic numbers in math. It is in the form as n = 2k + 1. Here integer is k. We can learn odd numbers by using this form. If we want to identify the odd numbers means divide that number by 2 and if we get the remainder, that number is odd.

The odd numbers are generated by using function as `(x(1 + x))/(1 + x)`  = x + 3x2 + 5x3 +......

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Learn odd numbers with example:


Problem 1: Find the odd numbers  from given set

{ 3, 6, 14, 7, 21, 34, 22, 47 }.

Answer:

The odd numbers are { 3, 7, 21, 47 }. Because these numbers are ended with 1,3, 7 and it does not divided by two.

Problem 2: Add and multiply the following odd numbers:

3 + 9 = 12.

53 + 7 = 60.

33 x 5 = 165.

Exercise problem for odd numbers:

Find the odd number from set { 190, 321, 453, 628 }.

Answer: { 321, 453 }.

Definition Reflexive Math

Introduction:

Reflexive is one of the properties in math. It is one of the postulates in equality. Reflexive property means anything equals to itself that is a = a where a is the set of numbers or number. Using reflexive property many geometric proofs were proved. Since it has equality we can find the answer for the given expressions.

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Example problems - Definition reflexive math:

Example 1 - Definition reflexive math:

Use the reflexive property to find the answer for x and the expressions are x+9 and 2x+8.

Solution:

The given expressions are x+9 and 2x+8

We the reflexive property anything equals to same thing

So, the expression can be written as x+9=2x+8

Add –x on both sides in the above equation

X+9+(-x) =2x+8+(-x)

X+9-x = 2x+8-x

9=x+8

Now subtract 8 on both sides

9-8=x+8-8

1=x

The value of x is 1.

Example 2 - Definition reflexive math:

Use the reflexive property to find the answer for y and the expressions are 2y+8 and y+8.

Solution:

The given expressions are 2y+8 and y+8.

We the reflexive property anything equals to same thing

So, the expression can be written as 2y+8 = y+8.

Add –y on both sides in the above equation

2y+8 +(-y)= y+8+(-y)

2y+8-y=y+8-y

y+8=8

Now subtract 8 on both sides

y+8-8=8-8

y=0

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More Example problems - Definition reflexive math:


Example 3 - Definition reflexive math:

Use the reflexive property to find the answer for x and the expressions are 3x+9 and 2x+8.

Solution:

The given expressions are 3x+9 and 2x+8

We the reflexive property anything equals to same thing

So, the expression can be written as 3x+9=2x+8

Add –xx on both sides in the above equation

3x+9+(-2x) =2x+8+(-2x)

3x+9-2x = 2x+8-2x

x+9 =8

Now subtract 9 on both sidesx

x+9-9=8-9

x=-1

The value of x is -1.

Example 4 - Definition reflexive math:

Use the reflexive property to find the answer for y and the expressions are 2y+8 and4y+8.

Solution:

The given expressions are 2y+8 and 4y+8.

We the reflexive property anything equals to same thing

So, the expression can be written as 2y+8 = 4y+8.

Add –2y on both sides in the above equation

2y+8 +(-2y)= 4y+8+(-2y)

8=2y+8

Now subtract 8 on both sides

8-8 =2y+8-8

2y=0

So, y=0.

Maximum Math Terms

Introduction to maximum math terms:
The math term maximum is define different kinds of versions. They are find out the maximum value from given data, determine the maximum distance between two points and the maximum value of set theory function. The greatest value is defined in math as maximum. The integration is used for find out the maximum value in set element.


Explanation for terms maximum in math


Procedures for finding the maximum value in math:

Step 1: Collect the set of data and use the comparison method.

Step 2: The comparison method is arrange the numbers in ascending order.

Step 3: At last find the maximum number from above order of data.

For example take  the set of data as 4, 2, 1, 6, 12, 3, 10. The comparison method is used for data arrangement in ascending order. The elements in ascending order is 1, 2, 3, 4, 6, 10, 12. The greatest value of given set of data is 12.

Maximum in other branch of math terms:

The set operations are used for finding the maximum value in functions. In math the optimization is determine the maxima from set of data and the real valued function is explain the global maximum. The bounded value theorem is essential one for global maximum determination.

In terms maxima, the expression A = { ai }N is explain the maxima for set of elements. The last element of this order is denote the maximum value of set.  The order statistics has different types and this is one type of statistics.

Consider the example as { 1, 5, 8, 4 } and arrange the set of data in ascending order as { 1, 4, 5, 8 }.Finally, the maximum value of set is 8.

Maximum in profit:

The term maximum in profit is how much of profit the company get.

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Examples for maximum in math terms


Problem 1:

Find the maximum value from the set of numbers 8, 4, 1, 20, 15, 9.

Answer:

Arrange the set of numbers in ascending order as 1, 4, 8, 9, 15, 20.

Set’s maximum value is 20.

Problem 2:

Find the maximum profit from P = 6x – 0.04x^2.

Answer:

If `(dP)/dx` = 0 then 6 – 0.08x = 0

`(d^2P)/(dx^2)` = -0.08 it has maximum profit.

If x = 200, profit is 400.

Excercise for maximum terms in mat:

1. Find out the maximum from given set of data { 12, 15, 11, 22, 9 }.

Answer: Maximum value of set is 22.

2. Find out the maximum value from set of numbers: 110, 120, 108, 115.

Answer: The minimum value is 120.

Video Tutorials for Math

Introduction about video tutorials for math:

Math is very interesting and complicated subject. We have various types of method and procedures to solve the math question. Tutorial is one of the techniques to improve the student’s knowledge. Students can learn their own speed, and their convenience. Interactive classes can do by a tutor during tutorials. One of the main benefits is, students clarify their doubts frankly as well as immediately. Here we are going to learn about some example problems of tutorials for math.

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Example problems of video tutorials for math:


Example 1:

A train takes to travels 200 miles in 4 hours. How long the train travels in 8 hours?

Solution:

Step 1: We know that the train travel 200 miles. If a train travel 8 hours how many miles the train travel?

Step 2: it’s the direct proportionality.

Step 3: We know that

Hour’s         miles

4               200

8                 x

Step 4: Now we need to solve the value of x using cross multiplication. So, we get 4x= 1600

Step 5: Divide using 4 on both the sides. So, the train travelled 400 miles.

Example 2:

Using these lines. Find the intersecting points?

y = 2y+2

y = x+12

Solution:

Step 1:

y = 2x+2      ------------> ( 1 )

y = x+12      -------------> ( 2 )

Step 2:

We know that the two lines are equal. So, it can be written as ,2x + 2 = x + 12

Step 3:  Here we need to solve the value of x. So, 2x – x = 12 – 2.Therefore, the value of x is 10

Step 4:

Substituting the value of x=10 in equation (1)

y = 2x + 2

= 2 ( 10 ) + 2

= 20 + 2   = 22

y = 22

The two given two lines intersect at a point (10, 22)

This is the example of intersecting lines.

These are the example problems of video tutorials for math .

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One more example of video tutorials for math:


Example 3:

The sum of the least and greatest of 3 consecutive integers is 160. What are the values of the 3 integers?

Solution:

Step 1: Let us assume the integers are x,x+1 and x+2 .

Step 2: And we know that , the least and greatest of the 3 consecutive integer is 160 .

Step 3: So, we can arrange the equation, that is x+(x+2) = 160 .

Step 4: Here we need to solve this equation. So 2x+2 = 160. (Subtract 2 on both the sides)

Step 5: Therefore x= 79. Therefore the least integer is 79.

Step 6: Substitute the x value is x+1, x+2 .

Step 7: Therefore, 79+1 = 80 and 79+2= 81.

Step 8: Therefore, the values of 3 integers are 79, 80 and 81.

These are the examples of video tutorials for math.

Math Skills Practice

Basic Math Definition:

Arithmetic is the one of the Basic mathematics definition. Arithmetic deals with the properties of numbers and ways of combining them through addition (+), subtraction (-), multiplication (*), and division (/) And basic math definition are contain with the arithmetic, Fractions, Decimals, Roots, Exponents, Factorials, Modular and Interval Arithmetic these are the contains with the basic math definition.

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Basic Math Definition Examples:


Arithmetic practice Examples:

Adding Arithmetic Example:

56977 + 21454

56977
+ 21454
---------
78431
---------
Subtracting Arithmetic Example:

782138 – 574121

782138
( - )    574121
----------
208017
----------
Multiplication Arithmetic Example:

875 x 69

875 x 69
-------------
7 8 7 5
5 2 5 0
-------------
6 0 37 5
-------------
Division Arithmetic Example:

45` -: ` 9

9) 45 (5
45
------
0
------

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Fractions practice Examples:


Fraction practice Example 1:

Add Two Fractions  ` 5/6 + 9/8`

Step 1:

For the denominators here, the 6 and 8, a common denominator for both is 24.

Step 2:

With the common denominator, the `5/6` becomes a `20/24 ` and the  `9/8 ` becomes a `27/24`

So, the problem here is to add `20/24` and `27/24`

Step 3:

The two denominators are same adding numerator values then ,we get  `47/24`

Step 4:

The fraction `47/24` is an improper fraction (the numerator is greater than the denominator).
Step 5:

The whole number part of the mixed number is found by dividing the 47 by the 24.

Step 6:

The fractional element of the mixed number is found by using the remainder of the division, which in this case is 23 (47 divided by 24 is 1 remainder 23).

Answer:

`1 23/24`

Reduce the fraction Practice Example:

Reduce the given factor 6/63

Solution:

The fraction `6/63` is not reduced to lowest terms. We can decrease this fraction to lowest terms by dividing both the numerator and denominator by 3.

Greatest Common Factor (GCF) of the numbers 6 and 63. So, this fraction reduced to lowest terms is `2/21`

Answer:  ` 2/21`

Subtracting the Fraction Practice example:

Subtracting the given fraction `8/5` and` 3/5`

Solution:

Step 1:

The given faction denominators (5) are equal.

Step 2:

Therefore the fraction, `5 /5` can be reduced further to 1, because anything over itself (other than zero) is 1.

Step 3:

The fraction 1 is not reduced to lowest terms. We can decrease this fraction to lowest terms by dividing both the numerator and denominator by 5.

Step 4:

Greatest Common Factor (GCF) of the numbers 0 and 5.

So, this fraction reduced to lowest terms is 1

Answer

The subtracting fraction is 1