Solving Equations: Basic Equations: Type 1: Example: 5x+2 = 3x + 6 => x = (6-2)/(5-3) = 4/2 = 2 Type 2: Example: (x+1)(x+2)= (x+3)(x+4) => x= (12-2)/(1+2-3-4) = 10/(-4) = -5/2 Type 3: Example: (3x+2)/(2x+1) = 4/3 x = (4-6)/(9-8) = -2/1 = -2     Type 4: Example: 1/(x+2) + 2/(x+1) = 0 => x = (-1-4)/3 = -5/3   Sunyam SamasyaSamuccaye Sanskrit Name:      शून्यं साम्यसमुच्च्ये English Translation: It has different meaning in different context. In general its meaning is related to equating with 0 (Sunya). First Meaning: A term which occurs as a common factor in all the terms is equated to 0. Examples: 12x+3x = 4x + 3x … As x is common factor both sides.So, x = 0. 9(x+3) = 4(x+3) … As (x+3) is common term both sides. So, x+3 = 0. x = -3. Second Meaning: The product of the independent terms is same both sides then equated to 0. Examples: (x+5)(x+4) = (x+2)(x+10) As product of independent terms (non-x terms): 5 x 4 = 2 x 10 , is same on both sides. Therefore, x=0. Third Meaning: The sum of the Denominators of two fractions having the same numerical numerator is equated to 0. Examples: 1/(2x-1) + 1/(4x-1) = 0 Therefore, sum of denominators: 2x-1 + 4x-1 = 0 On solving x = 1/3 Fourth Meaning: The sum of the Numerators and the sum of the Denominators is the same, then that sum equated to 0. Examples: (2x +9)/ (2x +7) = (2x +7)/ (2x +9) Here, Addition of both numerators = Addition of both Denominators. Thus, 2x + 9 + 2x + 7 = 0 Hence 4x + 16 = 0 hence x = -4 If there is a numerical factor in the algebraic sum, then we remove that factor (3x +4)/ (6x +7) = (x +1)/ (2x +3) Here, Addition of both numerators = 4x +5 Addition of both Denominators = 8x + 10 =2(4x +5) where, 2 is the numerical factor. So remove it 4x +5 =0. Hence x= -5/4 NOTE: In above both examples, when we do cross multiplication the x2 term is getting cancelled. So instead of being quadratic eq … [Read more...]