answer:To solve this problem, we need to determine the value of x, which represents the number of TV episodes that Emily watched. Let's break down the information given: Length of the flight: 10 hours Length of each TV episode: 25 minutes Length of sleep: 4 and a half hours = 4.5 * 60 = 270 minutes Length of each movie: 1 hour and 45 minutes = 1.75 * 60 = 105 minutes Total time spent on TV episodes: x * 25 minutes Total time spent on sleep: 270 minutes Total time spent on movies: 2 * 105 minutes Total time spent on activities: x * 25 + 270 + 2 * 105 minutes Total length of the flight: 10 hours = 10 * 60 = 600 minutes We can set up the equation as follows: Total length of the flight - Total time spent on activities = Remaining time in the flight 600 - (x * 25 + 270 + 2 * 105) = 45 Let's simplify and solve for x: 600 - (25x + 270 + 210) = 45 600 - (25x + 480) = 45 To isolate x, we subtract 600 from both sides of the equation: 600 - 600 - (25x + 480) = 45 - 600 -25x - 480 = -555 To solve for x, we divide both sides of the equation by -25: (-25x - 480) / -25 = -555 / -25 x + 19.2 = 22.2 To isolate x, we subtract 19.2 from both sides of the equation: x + 19.2 - 19.2 = 22.2 - 19.2 x = 3 The value of x is 3. 3 The answer is: 3

answer:We want to find the value of the number in the given situation. We are given that eighty-five more than the square of a number is the same as the square of the quantity that is X less than the number. We can express this information in the form of an equation. Let's call the number "n". According to the given information, we can set up the equation: n^2 + 85 = (n - X)^2 To solve for the value of X, we can simplify the equation and solve for X. Expanding the right side of the equation, we have: n^2 + 85 = n^2 - 2nX + X^2 Subtracting n^2 from both sides of the equation, we get: 85 = -2nX + X^2 Rearranging the equation, we have: X^2 - 2nX + 85 = 0 Now, we can solve this quadratic equation for X. Given that the value of the number is 6, we can substitute it into the equation: X^2 - 2(6)X + 85 = 0 Simplifying, we have: X^2 - 12X + 85 = 0 To factor this equation, we need to find two numbers that multiply to 85 and add up to -12. After factoring, we find that the equation can be written as: (X - 5)(X - 17) = 0 Setting each factor equal to zero, we have: X - 5 = 0 or X - 17 = 0 Solving each equation, we find: X = 5 or X = 17 Since we are looking for the value of X, we take the solution X = 17. The value of X is 17. The answer is: 17

answer:If frac{n+6}{n} is an integer, then n+6 must be divisible by n. This means that the remainder when n+6 is divided by n is 0. Using the Remainder Theorem, we can write n+6 as a polynomial in terms of n: n+6=ncdot1+6. So, n must be a factor of 6. The positive integral factors of 6 are 1, 2, 3, and 6. The sum of these values is 1+2+3+6=boxed{12}. The answer is: 12

answer:Substituting x = -5 into the equation x^2 = y - 3, we get (-5)^2 = y - 3. Simplifying, we have 25 = y - 3. Adding 3 to both sides, we find y = 28. Therefore, the value of y is boxed{28}. The answer is: 28