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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Step 1: Solve the equation 3x = 2x + 18. 3x - 2x = 18 x = 18 Check: 3(18) = 2(18) + 18 54 = 36 + 18 54 = 54. ✅ The solution is x = 18. Step 2: Solve the equation 5t - 3 = 3t - 5. 5t - 3t = -5 + 3 2t = -2 t = (-2)/(2) t = -1 Check: 5(-1) - 3 = 3(-1) - 5 -5 - 3 = -3 - 5 -8 = -8. ✅ The solution is t = -1. Step 3: Solve the equation 5x + 9 = 5 + 3x. 5x - 3x = 5 - 9 2x = -4 x = (-4)/(2) x = -2 Check: 5(-2) + 9 = 5 + 3(-2) -10 + 9 = 5 - 6 -1 = -1. ✅ The solution is x = -2. Step 4: Solve the equation 4z + 3 = 6 + 2z. 4z - 2z = 6 - 3 2z = 3 z = (3)/(2) Check: 4((3)/(2)) + 3 = 6 + 2((3)/(2)) 6 + 3 = 6 + 3 9 = 9. ✅ The solution is z = (3)/(2). Step 5: Solve the equation 2x - 1 = 14 - x. 2x + x = 14 + 1 3x = 15 x = (15)/(3) x = 5 Check: 2(5) - 1 = 14 - 5 10 - 1 = 9 9 = 9. ✅ The solution is x = 5. Step 6: Solve the equation 8x + 4 = 3(x - 1) + 7. 8x + 4 = 3x - 3 + 7 8x + 4 = 3x + 4 8x - 3x = 4 - 4 5x = 0 x = (0)/(5) x = 0 Check: 8(0) + 4 = 3(0 - 1) + 7 0 + 4 = 3(-1) + 7 4 = -3 + 7 4 = 4. ✅ The solution is x = 0. Step 7: Solve the equation x = (4)/(5)(x + 10). 5x = 4(x + 10) 5x = 4x + 40 5x - 4x = 40 x = 40 Check: 40 = (4)/(5)(40 + 10) 40 = (4)/(5)(50) 40 = 4 × 10 40 = 40. ✅ The solution is x = 40. Step 8: Solve the equation (2x)/(3) + 1 = (7x)/(15) + 3. Multiply the entire equation by 15 (the least common multiple of 3 and 15). 15((2x)/(3)) + 15(1) = 15((7x)/(15)) + 15(3) 10x + 15 = 7x + 45 10x - 7x = 45 - 15 3x = 30 x = (30)/(3) x = 10 Check: (2(10))/(3) + 1 = (7(10))/(15) + 3 (20)/(3) + 1 = (70)/(15) + 3 (20)/(3) + (3)/(3) = (14)/(3) + (9)/(3) (23)/(3) = (23)/(3). ✅ The solution is x = 10. Step 9: Solve the equation 2y + (5)/(3) = (26)/(3) - y. 2y + y = (26)/(3) - (5)/(3) 3y = (21)/(3) 3y = 7 y = (7)/(3) Check: 2((7)/(3)) + (5)/(3) = (26)/(3) - (7)/(3) (14)/(3) + (5)/(3) = (19)/(3) (19)/(3) = (19)/(3). ✅ The solution is y = (7)/(3). Step 10: Solve the equation 3m = 5m - (8)/(5). 3m - 5m = -(8)/(5) -2m = -(8)/(5) m = (-8)/(5)-2 m = (8)/(10) m = (4)/(5) Check: 3((4)/(5)) = 5((4)/(5)) - (8)/(5) (12)/(5) = (20)/(5) - (8)/(5) (12)/(5) = (12)/(5). ✅ The solution is m = (4)/(5).