Among the equations examined, only "65 - 2x = 6x + 9" has the solution set {7}. The others lack {7} in their solutions.
Let's analyze each equation to determine if the solution set includes the element {7}.
3 + 4x = 5x - 4: Combine like terms and solve for x: 3 + 4x = 5x - 4 → 7 = x - 4 → x = 11. The solution set is not {7}, so this equation does not have {7} as its solution.
65 - 2x = 6x + 9: Combine like terms and solve for x: 65 - 2x = 6x + 9 → 56 = 8x → x = 7. The solution set is {7}, so this equation has {7} as its solution.
x^2 + 1 = 8x - 5: Rearrange and solve for x: x^2 - 8x + 6 = 0. Using the quadratic formula, x = (8 ± √(8^2 - 4(1)(6))) / (2(1)). There are two solutions, but none of them is 7.
9 - x = 14 / x: Solve for x: 9 - x = 14 / x → x^2 - 9x + 14 = 0. Using the quadratic formula again, x = (9 ± √(9^2 - 4(1)(14))) / (2(1)). There are two solutions, and neither is 7.
Therefore, only equation 2 (65 - 2x = 6x + 9) has {7} as its solution set. The summary is that among the provided equations, only one has the solution set {7}.