Respuesta :
Respuesta:
Solving the first equation: [x + (x+5)(x-3) = 3 + x(x+1)] First, distribute the binomial ((x+5)(x-3)): [x + (x^2 + 2x - 15) = 3 + x^2 + x] Combine like terms on both sides: [x + x^2 + 2x - 15 = 3 + x^2 + x] Rearrange the equation: [x^2 + 2x - 15 = 3 + x^2 + x] Subtract (x^2) from both sides: [2x - 15 = 3 + x] Now, isolate the variable (x): [2x - x = 3 + 15] [x = 18]
Solving the second equation: [\frac{3x}{2} - \frac{5}{3} = x - 1] To get rid of the fractions, multiply both sides of the equation by the least common multiple (LCM) of 2 and 3, which is 6: [6\left(\frac{3x}{2}\right) - 6\left(\frac{5}{3}\right) = 6(x - 1)] Simplify: [9x - 10 = 6x - 6] Rearrange and solve for (x): [9x - 6x = 10 - 6] [3x = 4] [x = \frac{4}{3}]
Solving the third equation: [\frac{x+2}{4} - \frac{x-4}{2} = 2] First, find a common denominator for the fractions: [2(x+2) - 4(x-4) = 8] Distribute and simplify: [2x + 4 - 4x + 16 = 8] Combine like terms: [-2x + 20 = 8] Rearrange and solve for (x): [-2x = 8 - 20] [-2x = -12] [x = 6]
Solving the fourth equation: [\frac{x-3}{5} = \frac{x+1}{3} - 2] Multiply both sides by 5 and 3 to eliminate the fractions: [3(x-3) = 5(x+1) - 30] Distribute and simplify: [3x - 9 = 5x + 5 - 30] Combine like terms: [3x - 9 = 5x - 25] Rearrange and solve for (x): [2x = 16] [x = 8]
Solving the fifth equation: [\frac{3x-1}{7} + 2x - \frac{1}{3} = x] Multiply both sides by 7 and 3 to eliminate the fractions: [3(3x-1) + 42x - 7 = 21x] Distribute and simplify: [9x - 3 + 42x - 7 = 21x] Combine like terms: [51x - 10 = 21x] Rearrange and solve for (x): [30x = 10] [x = \frac{1}{3}]