Steps  x + y = 9 and xy = 14  
1.  Find a pattern in the given parameters, and arrange them in a similar pattern.  In this case, x + y = 9 xy = 14 

2.  Observe carefully to assume combinations that fit the pattern.  In this case, for xy = 14 14 may be written as 2 × 7, and 2 × 7, other than 1 × 14 So, possible combinations are {x,y} = {2,7}, {2,7}, {1,14}, {1,14} 

3.  Test the combinations, to finalize assumptions.  In this case, testing the combinations in: x + y = 9 {x,y} = {2,7} gives: 2 + 7 = 9 This assumption is finalized. {x,y} = {2,7} gives: 2  7 = 9 ≠ 9, and thus discarded. Similarly, {x,y} = {1,14} and {1,14} are discarded. 

4.  Find other values, if required, using the finalized combinations.  Using the finalized assumption {x,y} = {2,7}, we get: Replacing x = 2, 2 + y = 9 gives y = 7 And, replacing x = 7 7 + y = 9 gives y = 2 Thus, the solutions are {x,y} = {2,7} or {7,2} 
5 may be written as 4 + 1, and thus, 5 4+1 4 1 1  =  =  +  = 2 +  2 2 2 2 2 By the above observation, x=1/2 will also satisfy the expression. Thus, for x + 1/x = 5/2, x = 2 or x = 1/2 
15 may be written as 3 × 5, 3 × 5, 1 × 15, 1 × 15 Using this assumption: Replacing x = 3, 3 5 9 + 25 34  +  =  =  5 3 15 15 This assumption is finalized. Similarly, of the other assumptions, only x = 5 is finalized. Thus, for the above equation, x = 3 or x = 5 Note that this being a Quadratic equation, after finalizing two assumptions, we do not need to test the others. 
Note that, 82 720 841121 2  =  =  319 319 319 This gives us, 5x+9 5x9 29 11  +  =    5x9 5x+9 11 29 Thus, (5x+9)/(5x9) = 29/11 or 11/29 Again, 29 = 20 + 9 = (5 × 4) + 9 And, 11 = 20  9 = (5 × 4) + 9 This observation in (5x+9)/(5x9) = 29/11 gives us x = 4 And, working the other option (in conventional method), gives us x = 81/100 
The mere observation of 2x+7 gives us x+x+3+4. Thus, 2x+7 may be written as (x+3) + (x+4), which transforms the given expression to: 2x+7 (x+3) + (x+4)  =  (x+3)(x+4) (x+3) (x+4) 1 1 =  +  x+3 x+4 
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