This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems: प्रश्न 1: 3x = 2x + 18 Step 1: Subtract 2x from both sides. 3x - 2x = 18 x = 18 The solution is x = 18. प्रश्न 2: 5t - 3 = 3t - 5 Step 1: Subtract 3t from both sides. 5t - 3t - 3 = -5 2t - 3 = -5 Step 2: Add 3 to both sides. 2t = -5 + 3 2t = -2 Step 3: Divide by 2. t = (-2)/(2) t = -1 The solution is t = -1. प्रश्न 4: 4z + 3 = 6 + 2z Step 1: Subtract 2z from both sides. 4z - 2z + 3 = 6 2z + 3 = 6 Step 2: Subtract 3 from both sides. 2z = 6 - 3 2z = 3 Step 3: Divide by 2. z = (3)/(2) The solution is z = (3)/(2). प्रश्न 5: 2x - 1 = 14 - x Step 1: Add x to both sides. 2x + x - 1 = 14 3x - 1 = 14 Step 2: Add 1 to both sides. 3x = 14 + 1 3x = 15 Step 3: Divide by 3. x = (15)/(3) x = 5 The solution is x = 5. प्रश्न 7: x = (4)/(5)(x + 10) Step 1: Distribute (4)/(5) on the right side. x = (4)/(5)x + (4)/(5) × 10 x = (4)/(5)x + 8 Step 2: Subtract (4)/(5)x from both sides. x - (4)/(5)x = 8 (5x - 4x)/(5) = 8 (1)/(5)x = 8 Step 3: Multiply by 5. x = 8 × 5 x = 40 The solution is x = 40. प्रश्न 8: (2x)/(3) + 1 = (7x)/(15) + 3 Step 1: Multiply the entire equation by the least common multiple (LCM) of the denominators (3 and 15), which is 15. 15 ( (2x)/(3) ) + 15(1) = 15 ( (7x)/(15) ) + 15(3) 5(2x) + 15 = 7x + 45 10x + 15 = 7x + 45 Step 2: Subtract 7x from both sides. 10x - 7x + 15 = 45 3x + 15 = 45 Step 3: Subtract 15 from both sides. 3x = 45 - 15 3x = 30 Step 4: Divide by 3. x = (30)/(3) x = 10 The solution is x = 10. प्रश्न 10: 3m = 5m - (8)/(5) Step 1: Subtract 5m from both sides. 3m - 5m = -(8)/(5) -2m = -(8)/(5) Step 2: Divide by -2. m = (-8)/(5)-2 m = (8)/(5 × 2) m = (8)/(10) Step 3: Simplify the fraction. m = (4)/(5) The solution is m = (4)/(5). उदाहरण 3: (6x+1)/(3) + 1 = (x-3)/(6) Step 1: Multiply both sides by the LCM of the denominators (3 and 6), which is 6. 6 ( (6x+1)/(3) + 1 ) = 6 ( (x-3)/(6) ) 2(6x+1) + 6(1) = x-3 Step 2: Distribute the 2 on the left side. 12x + 2 + 6 = x - 3 12x + 8 = x - 3 Step 3: Subtract x from both sides. 12x - x + 8 = -3 11x + 8 = -3 Step 4: Subtract 8 from both sides. 11x = -3 - 8 11x = -11 Step 5: Divide by 11. x = (-11)/(11) x = -1 The solution is x = -1.