Equations with no Solution

Friends, Previously we have discussed about area of an octagon calculator and today we are going to discuss about the linear equations with no solution which is a part of ap state board of secondary education. Before I start first I will tell you about what is linear equation and solving linear equations. A linear equation is an algebraic equation which consists of a term that can be a single variable and constant or product of constants. A linear equation may have more than one variable.
A linear equation can have no solution. For instance:
0x + 2 = 5; here in this given linear equation since 0 is multiplied by x so this will result in a 0. And 0 + 2 is not equals to 5, so from this we can say that any linear equation has a zero as the coefficient of x will not give any solution. Such equations are known as Equations with no Solution.
Whenever we start solving an equation, we always try to make its simplest form. We always start with an assumption that the equation is having an actual solution. We try to solve it and when we end it with an uncertain and wrong answer then it indicates that our assumption was wrong and the equation has no solution. So the statement 2 = 5 in above example is false.
Let’s take one more example: Solve the equation      12 + 5x – 9 = 7x + 5 – 3x
Solution: 12 + 5x – 9 = 7x + 5 – 3x       (given equation)
12 -9 – 5 = 7x –2 x – 5x          (simplify it)
-2 = 0  (Answer)
So the above example gave the result -2 = 0 that is not true; that means that this equation has no solution.(want to Learn more about Equations, click here),
So such equations which has 0 as a coefficient of the variable or the equations which give a wrong or false statement are always the Equations with no Solution.
In the next session we are going to discuss Equations with no Solution
and Read more maths topics of different grades such as Compound Inequalities
in the upcoming sessions here.

Rationalizing the Denominator

Previously we have discussed about adding scientific notation calculator and In today's session we are going to discuss about Rationalizing the Denominator which comes under andhra pradesh state board of secondary education,  The process of eliminating a square root or imaginary number from the denominator of a fraction is termed as rationalization. A rational number is a number that can be expressed as the ratio of two integers, such as 2/3. Any number with a terminating decimal part is a rational number. If decimal part begins to repeat in a numeral, it is also rational number, such as .0303030..., since this can be expressed as 1/33.  Numbers that are not rational are known as irrational.  Examples of irrational numbers are the square root of 2, pi, and e. Examples of rational no are 2/4,3/5.

Rationalizing the Denominator is done to simplifying fractions into simplest form i.e. the denominator should not have an irrational number or a complex number.
In case of real number there are 3 cases where we use rationalization
1. The denominator having single square root
2. The denominator having single higher root
3. The denominator having sums and differences of square roots
case 1:
 When you have a single square root in the denominator you just multiply top and bottom by it.
Example:

1/
=1*/*

=1*/3
in th above example  the denominator is . so we need to rartionaze it forthis we multiply and divide it by the same no . doing this the value of the expression remains same and the square root gets removed from the denominator as * gives 3 which is not irrational

case 2:When we have denominators having higher root
example:   1/

=1/(*)

 = 1/a
since this is a 3rd root, in order to remove the root from the denominator we have to get cube of the values inside the root. To get this we multiply the denominator and numerator with the cube root of the square of the value. Finally we get the cube root of the cube of the value and thus the cube root is removed and we get a rational no in the denominator. (Know more about Rationalizing the Denominator in broad manner, here,)
CASE 3: When the denominator is having   sums and differences of square roots
example:2/1-

=2*(1+)/(1-)(1+)

=2*(1+)/1-3

=2*(1+)/-2

=-(1+)

In this case if a sum is in the denominator we  multiply the denominator and numerator with the difference. And if we have a difference in the denominator we multiply the denominator and numerator with the sum. Thus we rationalize denominator.
In The Next Session We Are Going to Discuss Equations with no Solution

and Read more maths topics of different grades such as Equations and Inequalities in the upcoming sessions here.    

Math Blog on Subtracting Rational Expressions

Previously we have discussed about length of an arc calculator and In this session I am going to tell you about the subtraction in rational expressions which comes under state board of secondary education andhra pradesh. Before starting the topic we must know about the rational numbers. The Rational numbers can be any numbers that can be represented in a form of fraction say a/b where a is the numerator that can be any integer and b is the denominator that is a whole or non zero number. For instance: 2/3, 4/5 etc.
While subtracting rational numbers, we need to convert denominators to a common denominator; otherwise we cannot solve the problem. Now here is the method by which we can solve such problems in mathematics:

1.    First find out the common denominator.

2.    To find common denominator, first find the LCM (lcm calculator) that is Least Common Multiple of all the denominators. We can use the listing method for this, where we can list all the multiples of all the denominators until we find a number that exist in all the lists.

3.    After finding LCD (Least Common Denominator), proceed to the subtraction.

4.    Now make the lowest term of the resultant term and put it into the answer.

We can understand this by solving problems related to the rational expressions subtractions. Here is an example for this.(want to Learn more about Rational Expressions, click here),
Subtract        8/10x – 2/5x
                   First take LCM of 10 and 5, so that we can get the LCD.
                   10: 2*5
                   5:  1*5
So the LCM is 2*5 that is 10. So 10 is also the common denominator.
(8x – (2*2)*x) / 10x (Subtract)
4x/10x (Simplify the term by dividing both the denominator and numerator by 2)
2/5 (answer)
One more example:
Subtract        [x/(x - 1)] – [1/x]    (Take LCM to get LCD).
                   [x*x – (x - 1)] / [(x - 1)*x](Simplify each term).
                   (X2 – x – 1) / (x2 - x)
In the next session, we will learn about Rationalizing the Denominator
and if anyone want to know about Solving Multi Step Inequalities then they can refer to Internet and text books for understanding it more precisely.

Simplifying Rational Expressions

Hi students,Previously we have discussed about area of ellipse calculator and now I am going to discuss about Simplifying Rational Expressions which is a part of ap state secondary education board. The number, which can be expressed in the form of a/b, where a and b are both integers and b cannot be equal to 0 are called as rational numbers. Here a is termed as a numerator and b is termed as a denominator. Any single integer can be written in a fractional form by Solving Equations with Fractions. (Know more about Rational Expressions in broad manner, here,)

In this session, we shall be discussing algebra simplifying rational expressions. Let us take some examples to understand how to simplify rational expressions.

Example 1:- The given expression is 3y / y² . How to simplifying the rational number?
Solution :-
step 1 :- 3*y / y*y
step 2 :- 3 / y
(this is a fraction expression. In this expression cancel the common value . For this expression numerator is 3y and denominator is y². Thus we cancel the common term 'y' ).
Any single digit can be written as a rational number but denominator should not not zero.
Example 2:- Simplify the rational number 12.
Solution :-12 is a rational number because 12 can be rewritten as 12/1. Here 12 is the numerator and 1 is denominator value in which 1 is not equal to 0 and both are positive integer we can say that 12 is a rational number. Also it in simplest form.
Rule for addition of rational numbers:
If we are given, p/q+r/s to simplify, we write (p s+qr) / qs and then solve.

Some examples of addition of fractions:
Example :- (a+6)/3 +(a-9)/6
to solve the rational numbers.
solution :- step 1:- (a+6)*(6)+(3)*(a-9)/ 3 * 6
step 2:-(6a+36+3a-27) / 18
Step 3:-9a+9/18


In the next topic we are going to discuss Math Blog on Subtracting Rational Expressions and Read more maths topics of different grades such as Solving inequalities by addition and subtractions  in the upcoming sessions here.

Multiplying Rational Expressions

Hi children! Previously we have discussed about equilateral triangle area calculator and Today we are going to learn about Multiplying Rational Expressions which is part of andhra pradesh secondary board.
Let’s recall rational numbers. The numbers which can be expressed in form of p / q , where p and q are integers are q is not 0 are called rational numbers. All mathematical operators, i.e., addition, subtraction, multiplication and division can be performed on rational numbers.  While multiplying rational expressions, we need to multiply numerator with the numerator and multiply the denominator with the denominator, unlike addition and subtraction, where we need to find the LCM (lcm calculator) of the denominators. With only simple multiplication we get the result.
Let us understand it more clearly with multiplying Rational Expressions example.
Solve 3/5 *2/7
In this case we observe that 3 and 2 are the numerators and 5 and 7 are the denominators. (Know more about Rational Expressions in broad manner, here,)
So we multiply 3 * 2 =6 and 5 * 7 = 35
= 6/35 Ans.
In the same way if we have three rational numbers say 2/5, 3/8 and 6/9 and we have to multiply three rational numbers then we get
   ( 2/5)* (3/8 ) * ( 6/9)
 We observe that 2, 3 and 6 are the numerators and 5 , 8 and 9 are the denominators
 So multiplying numerators and denominators we get:
 = ( 2 * 3 * 6 ) / ( 5 * 8 * 9 )
= 36 / 360
 Now converting them to lowest terms we divide both numerator and denominator by 36 and get
= (36÷ 10 )/ (360 ÷10)
 = 1 / 10 Ans.
We should also remember that when we multiply any rational number by 1, the number remains unchanged and when the rational number is multiplied by 0 the product is always 0.
In the next topic we are going to discuss Simplifying Rational Expressions
and if anyone want to know about Solving Inequalities with Rational Numbers then they can refer to Internet and text books for understanding it more precisely.

Addition of Rational Expression

Numbers which can be expressed in form of p / q where p and q are integers and q < > 0 are called rational numbers.  Rational expressions are the  expressions which consists of rational numbers. Various mathematical Operation on Rational Expression can be done like addition, subtraction, multiplication and division. We can do all operations on Rational numbers.
On doing addition of Rational Expressions, the output is also a rational number. In order to do Addition of Rational Expression, we first need to make the denominators of the operands same. For this we will first take the L.C.M. of all the denominators given in the expression. Then we try to replace all the denominators with the L.C.M.
For this we multiply both numerator and the denominator with some of the factors of the denominator, so that the denominator is the L.C.M. now. Then we proceed to the addition of the numerators, keeping the denominators same.
Let us look at the example:
Solve the given Rational Expression:   2 +  4/5  + (-3/10)
 Here we observe the three denominators are 1, 5 and 10. L.C.M. of these three numbers is 10.
So to make the denominator of 2 as 10, we multiply num. and denominator by 10
To make the denominator of  (4/5 ) as 10, we multiply both numerator and denominator by 2 and
 ( -3 / 10 ) already has 10 as the denominator.
Now we proceed:
 = 2 +  4/5  + (-3/10)
 = 2 *10/10   + ( 4 * 2 ) / (5 * 2 )   + ( -3 /10)
= 20 /10   + 8 /10 + (-3 /10 )
 = ( 20 + 8 -3 ) / 10
= 25 /10
= 5 / 2 Ans.

This is all about addition of rational numbers. In the nest article we will deal with Division of Rational Expressions.

Division of Rational Expressions

Rational numbers can be written in the form of p / q where p and q are integers and q <> 0. rational expressions is the series of rational numbers combined together with mathematical operators.  We can perform all mathematical operations namely addition, subtraction, multiplication and division on rational expressions and it comes under andhra pradesh board of secondary education. Division of Rational Expressions is very simple and follow a set pattern of steps. Before we start dividing rational expression, we will first understand the concept of reciprocal of rational number. By reciprocal of a rational number, we mean dividing 1 by a rational number. For instance:
  reciprocal of 4 / 7  = 1/ (4/7)
                                     = 7/4.
Always remember there is no reciprocal of zero (0) and reciprocal of 1 is always 1.
While dividing rational expression, we first  write the reciprocal of divident and replace divide sign by multiplication
Further, the sum changes into simple sum of multiplication and the numerator is multiplied by numerator, denominator by a denominator.
eg:  (-3/5) ÷ (4/10)
 we see that reciprocal of 4/10 is 10 / 4, so the above sum becomes :
= (-3/5) * (10/4)
= ( -3 * 10 ) / ( 5* 4)
= -30 /20
converting into lowest term we get
= -3 / 2 Ans.
Like division there are many rational expressions applications, Sometimes division of Rational Expression becomes the part of the expression with other operators. In such cases, we follow the rule of DMAS, where
D- stands for Division,
M- stands for Multiplication
A- stands for subtraction
S- stands for subtraction .
It means that division is the first mathematical operation to be followed in the given expression which is to be simplified.(want to Learn more about Rational Expressions,click here),
eg (-2/5) ÷ (2/6) * (3/7)
 First we solve (-2/5) ÷ (2/6)
                     = (-2/ 5) * (6/2 )
                     = (-2 * 6 ) / ( 5* 2)
                     = -12/10
 Ptting this in given equation we get:
     = ( -12/10) * ( 3/7)
= (-12 * 3) / (10 * 7)
= -36 /70
= -18/ 35 Ans.
This is all about the Division of Rational Expressions  and if anyone want to know about Simplifying Rational Expressions then they can refer to Internet and text books for understanding it more precisely.Read more maths topics of different grades such as Compound Inequalities in the next session here.