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!