LINEAR EQUATIONS OF TWO VARIABLES-IMPORTANT TIPS

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IMPORTANT TIPS:

• ax+b = 0 (a,b are real numbers and a is nonzero) is a linear equation of one variable. Thus, 2x+3 = 0 and 0.4x-5 = 0 are linear equations of one variable.


• ax+by+c = 0(a,b,c are real numbers and a and b are not equal to zero simultaneously)is a linear equation of two variables. Thus, 3x-2y+4 = 0 and(2x/3)+(4y/5)-6 = 0 are linear equations of two variables.


• y = (-a/b)x-(c/b) or y = (-ax-c)/b is called y-form of the equation ax+by+c = 0


• If we have only one equation, ax+by+c = 0, then it has got infinite solutions because for each value of x we get some value of y which satisfies the equation.
  


• If we have a pair of equations a₁x+b₁y+c₁ = 0 and a₂x+b₂y+c₂ = 0,then there are three possibilities:
  
  1. The equations have a particular(unique) solution where unique values of x and y satisfy both equations. This happens when a₁b₂-a₂b₁≠ 0
  
  
  2. The equations do not have a solution, i.e. there is no set of values of both x and y which can satisfy both equations. This happens when a₁b₂-a₂b₁ = 0 and either b₁c₂-b₂c₁≠ 0 or c₁a₂-c₂a₁≠ 0
  
  
  3. The pair of equations has infinite solutions.This happens when a₁b₂-a₂b₁ = b₁c₂-b₂c₁ = c₁a₂-c₂a₁ = 0
     
  
  
  
  
• METHOD OF CROSS MULTIPLICATION:
  
  Let the given pair of equations be:
  
  a₁x+b₁y+c₁ = 0
  a₂x+b₂y+c₂ = 0
  
  (a₁, b₁ and a₂, b₂ are not zero simultaneously)
  
  To understand the procedure READ THE DETAILS IN THE FOLLOWING IMAGE and
  To read the details clearly ENLARGE IT by opening it in another window.
  
  
  
  
  
  What we have done while going from step (A) to step (B) is :
  
  Write the product of b₁ andc₂ (downward arrow) minus the product of b₂ and c₁ (upward arrow) below x. Follow the same procedure for y and 1.
  
  So, we get (B). NEXT ?
  

  x = b₁c₂ - b₂c₁
        a₁b₂ - a₂b₁
  
  and
  
  y = c₁a₂ - c₂a₁
        a₁b₂ - a₂b₁

  
  
  Thus, we get the values of x and y which satisfy the given pair of equations.
  


• PRACTICAL PROBLEMS:
  
  In solving practical problems, one must not forget that it is the lack of proper understanding of the language that makes the solution difficult.So, while solving practical problems with the help of linear equations :
  
  1. Read the statement carefully.
  2. Analyse it thoroughly.
  3. Form correct equations.
  
  
  This requires great patience and exhaustive practice.More you practise, more you understand the method.
  
  

DESSERT

2 is the only prime which is even!

INTRODUCTORY NOTE

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It is a general impression among highschool students and their parents that Mathematics is a very tough subject to digest.In fact, about 40% of a student's study time is utilized for mathematics only.It is obvious and essential because mathematics is not a novel or a collection of short stories which you finish in one go. It is a subject which asks for visualization, computing skills,power of analysis and persistence in effort.

MATHEMATICS differs from other subjects in the sense that you cannot afford to forget basic concepts and methods you have studied previously.You can and sometimes you do forget the story of a king and his adventures which you were tought in Class VI or Class VII. You cannot afford to forget Methods of Factorization or Mathematical Identities when you enter Class X.The story of Class VII has no connection to the one you are going to study in Class X, but the topics of mathematics do have basic connections as you move forward. This aspect makes Mathematics an entirely different subject.

REMEMBER:

MATHEMATICS, AS FAR AS I HAVE EXPERIENCED AND ENJOYED, IS NOT A TOUGH SUBJECT AT ALL

WHAT YOU MUST REMEMBER

The following is a list of some important mathematical rules, formulas, methods, etc. ,which you have already studied uptil now and which you cannot afford to forget.Most of the students ignore this very important fact and it is too late when they realise what they missed.You must remember:

• Tables of 1-20(at least)


• Squares of 1-25(at least)


• Cubes of 1-12


• Decimal Calculations


• Percentage Calculations


• Calculation of HCF/GCF and LCM


• Conversion of Units(e.g. 1 m= 100 cm)


• All important Mathematical Identities and their Proper Use.


• All methods of factorization


• Rules of Indices


• Rules and Methods connected to Ratio and Proportion


• All Geometrical Definitions and Concepts(point,ray,line.angle,plane,etc.)


• Types of Triangles and Quadrilaterals


• Properties of Triangles and Quadrilaterals


• Formulas to calculate Perimeter, Area, Volume,etc.


• Preliminary Trigonometry


• Preliminary Set Theory and Set Operations.

Once you have mastered these all, your studies will become much easier and more interesting.

A PUZZLE FOR FUN

Harsh went to a departmental store to purchase something.He spent half of the total amount he had when he entered the store.When he came out he had just as many paise as he had rupees while entering the store and half as many rupees as he had paise while entering the store.What amount actually Harsh carried when he entered the store?

WELCOME TO MY MATHS BLOG

Welcome my dear students,
Your MANOJSIR is now available on net to solve your problems of mathematics.I will be posting topics of mathematics of Class 10 of Gujarat Secondary Education Board(GSEB).I hope that this will be of help to every student of Class 10 studying in any EDUCATION BOARD OF INDIA.

This is my humble effort to serve my students.

INDEX-TOPIC SEARCH
*INTRODUCTORY NOTE

*LINEAR EQUATIONS OF TWO VARIABLES-TIPS

*HCF AND LCM OF POLYNOMIALS-TIPS

*RATIONAL EXPRESSION

*QUADRATIC EQUATION-TIPS

*ARITHMETIC PROGRESSION-TIPS

*INSTALMENTS-TIPS

*INCOME TAX-TIPS

*MEAN-TIPS

*SIMILAR TRIANGLES-TIPS

*CONDITIONS FOR SIMILARITY-TIPS

*SIMILARITY AND PYTHAGORAS' THEOREM-TIPS