answer:Let's assume the number of girls at the camp is G. The number of boys at the camp is G+33, as the number of boys exceeds the number of girls by 33. The total number of people at the camp is G + (G+33) = 133 Combining like terms, we get 2G + 33 = 133 Subtracting 33 from both sides, we get 2G = 100 Dividing both sides by 2, we get G = 50 Therefore, there were 50 girls present at the camp. 50 The answer is: 50

answer:To solve this problem, we need to determine the value of x, which represents the price of the primer per gallon before the discount. Let's break down the information given: Number of rooms: 5 Amount of primer needed per room: 1 gallon Amount of paint needed per room: 1 gallon Price of primer per gallon before the discount: x Discount on primer: 20% Price of paint per gallon: 25.00 Total cost of paint and primer: 245.00 We can set up the equation as follows: (Number of rooms * Price of primer per gallon * Amount of primer needed per room * (1 - Discount on primer)) + (Number of rooms * Price of paint per gallon * Amount of paint needed per room) = Total cost of paint and primer (5 * x * 1 * (1 - 20%)) + (5 * 25.00 * 1) = 245.00 Let's simplify and solve for x: 5x * 0.8 + 5 * 25.00 = 245.00 4x + 125.00 = 245.00 To isolate x, we subtract 125.00 from both sides of the equation: 4x + 125.00 - 125.00 = 245.00 - 125.00 4x = 120.00 To solve for x, we divide both sides of the equation by 4: x = 120.00 / 4 x = 30.00 The value of x is 30.00. 30 The answer is: 30

answer:For the equation to have two distinct integer solutions, the discriminant must be a perfect square. The discriminant is (-k)^2 - 4(2)(8) = k^2 - 64. We want k^2 - 64 to be a perfect square, so let's express it as (k-n)(k+n), where n is a positive integer. We have k^2 - 64 = (k-n)(k+n), so k^2 - (k-n)(k+n) = 64, and expanding, we get 2kn - n^2 = 64. Since 2kn - n^2 is an integer, n must be a divisor of 64. The divisors of 64 are pm 1, pm 2, pm 4, pm 8, pm 16, pm 32, pm 64. We can try each of these values for n to find the corresponding values of k. For example, when n=1, we have 2k-1=64, so k=32.5, which is not an integer. When n=2, we have 4k-4=64, so k=17. Continuing in this manner, we find the possible values of k are -33,-18,-10,10,18,33. The sum of these values is -33-18-10+10+18+33=boxed{0}. The answer is: 0

answer:We want to find the value of X in the given situation. To add two matrices together, we simply add their corresponding elements. For the given matrices, we have: begin{pmatrix} X & 0 1 & 2 end{pmatrix} + begin{pmatrix} -5 & -7 4 & -9 end{pmatrix} = begin{pmatrix} -2 & -7 5 & -7 end{pmatrix} Comparing the corresponding elements of the two matrices, we can set up the following equations: X - 5 = -2 0 - 7 = -7 1 + 4 = 5 2 - 9 = -7 Solving these equations, we find: X = 3 The value of X is 3. The answer is: 3