Equations reducible to a linear equations in two variables.

By Md. Bilal

Solution:    
Q.No(2)      2                 -        Not linear

Q.No(3)      2                 -        Not linear
Q .No(4)     2                 -        Not linear

Let’s think!

In the above table the equations are not linear. Can you convert the equations into linear equations?

Let’s remember!
We can create new variables making a proper change in the given variables.       Substituting the new variables in the given non-linear equations, we can convert them in linear equations. Also remember that the denominator of any fraction of the form cannot be zero.

Now see an example of non linear simultaneous equation to solve by elimination method.

Ex:
                ;    

Solution: 

          
        

            

------I

 --------II

               
    
Replacing  in equation I and II

4M + 5N = 7----III

3M + 4N =  5----IV

To solving equation III and IV  multiply eq.III by 3 and eq Iv by 4 we get 
12M + 15N = 21  ---A     

12M + 16N = 20 -----B

subtracting equation A and B
12M + 15N = 21       

12M + 16N = 20
-        -            -
一一一一一一一
  0    -   1N     =   1 
    
         ∴  N = -1

 put  N= -1 in equation III we get
4M + 5(-1) = 7
4M -5 =7
4M = 12
∴  M = 3

M=3 and N =-1

Now if we solve same equations by cramers rule then also we can get same solution.

Ex :
                    ----I

                   ------II

 D =    
                         D =  (4) X (4) -(3) X (5)
                     
                      D = 16 - 15
                          
                      D =  1

Dx  =    
                   Dx  =  (7) X (4) - (5) X (5)
                   
                   Dx  = 28-25 

                   Dx  = 3 

Dy  =     
                      Dy =  (4) X (5) - (3) X (7)
                     
                      Dy = 20 - 21 

                      Dy = - 1

Now here, (as variables are in      form).

But we want the values of X and Y so,

Taking reciprocals

     

As you can see we have solved same simultaneous equation by elimination method as well as cramers rule and  we got same values of X and Y  .
Now you try to solve questions of your exercise with both the methods and comment me whether it is possible to you or not ? also which method you find more easy? comment me in comment section.