answer:The line is given by [frac{x - 2}{4} = frac{y - 1}{-1} = frac{z + 3}{5}.] This means the direction vector of the line is (mathbf{d} = (4, -1, 5)). From the equation, (frac{x - 2}{4} = frac{y - 1}{-1},) [x - 2 = 4t, quad y - 1 = -t quad text{for some parameter } t.] Substitute (t = y - 1) into (x - 2 = 4t,) [x = 4(y - 1) + 2 = 4y - 2.] So, (x - 4y + 2 = 0). From (frac{y - 1}{-1} = frac{z + 3}{5},) [y - 1 = -5s, quad z + 3 = 5s quad text{for some parameter } s.] Substitute (s = frac{y - 1}{-5}) into (z + 3 = 5s,) [z = 5left(frac{y - 1}{-5}right) - 3 = -y + 2.] Thus, (y + z - 2 = 0). Using the point (1,4,-5): Substitute into the equations (x - 4y + 2 = 0) and (y + z - 2 = 0) to verify: [1 - 4(4) + 2 = -15 neq 0,] [4 - 5 - 2 = -3 neq 0.] Both conditions are not satisfied, indicating a mistake in using the incorrect relationships. Instead, compute the normal vector (mathbf{n}) directly using the cross product: The normal vector of the plane can be found as the cross product of any two perpendicular vectors on the plane, but since we have only one direction vector, we can't directly compute it without another vector. Suppose (t = 1) in the direction vector equation (x - 2 = 4t,) giving ((6, 0, -2)). Now, take the cross product of ((1, 4, -5)) and ((6, 0, -2)): [mathbf{n} = (1, 4, -5) times (6, 0, -2) = (4cdot(-2) - (-5)cdot0, (-5)cdot6 - (1)cdot(-2), 1cdot0 - 4cdot6) = (-8, -28, -24).] Finding (gcd(-8, -28, -24)) gives 4, so dividing by -4: [mathbf{n} = (2, 7, 6).] The equation of the plane using point (1,4,-5): [2(x-1) + 7(y-4) + 6(z+5) = 0] [2x + 7y + 6z - 8 - 28 - 30 = 0] [2x + 7y + 6z - 66 = 0] [x + frac{7}{2}y + 3z - 33 = 0] The equation in integer form: [2x + 7y + 6z - 66 = 0] Conclusion with boxed answer: [boxed{2x + 7y + 6z - 66 = 0}]

answer:1. **Combine the given equations:** [ begin{align*} E_1^2 - E_2^2 - E_3^2 + E_1E_2 &= 5040, E_1^2 + 2E_2^2 + 2E_3^2 - 2E_1E_2 - E_1E_3 - E_2E_3 &= -4968. end{align*} ] Adding these equations, we get: [ 2E_1^2 + E_2^2 + E_3^2 - E_1E_2 - E_1E_3 - E_2E_3 = 72. ] Simplifying, we have: [ E_1^2 + frac{E_2^2 + E_3^2 - E_1E_2 - E_1E_3 - E_2E_3}{2} = 36. ] 2. **Rearrange and factor the equation:** [ (E_1-E_2)^2 + (E_1-E_3)^2 + (E_2-E_3)^2 = 72. ] The possible sets of squares that sum to 72 are (49, 16, 7) and (64, 4, 4). 3. **Assign values to differences and solve:** - **Case 1:** (E_1-E_2)^2 = 49, (E_1-E_3)^2 = 16, (E_2-E_3)^2 = 7. [ E_1 - E_2 = 7, quad E_1 - E_3 = 4, quad E_2 - E_3 = pm sqrt{7}. ] This leads to non-integer values for E_2 and E_3, which contradicts the integer condition. - **Case 2:** (E_1-E_2)^2 = 64, (E_1-E_3)^2 = 4, (E_2-E_3)^2 = 4. [ E_1 - E_2 = 8, quad E_1 - E_3 = 2, quad E_2 - E_3 = 2. ] Solving these, we get E_1 = E_2 + 8, E_1 = E_3 + 2, E_2 = E_3 + 2. Thus, E_1 = 12. 4. **Conclusion with boxed answer:** The energy E_1 of the first particle is 12. The final answer is boxed{C}

answer:To solve this, we need to find a number that when multiplied by (frac{1}{15}) gives 9. The equation to solve is: [ ? times frac{1}{15} = 9 ] First, reverse this multiplication by dividing: [ 9 div frac{1}{15} = 9 times frac{15}{1} = 135 ] Thus, the original price of the potion was (boxed{135}).

answer:Let's call the number we're looking for "x". According to the problem, 20% of x is equal to 200. We can write this as: 0.20 * x = 200 To find x, we divide both sides of the equation by 0.20: x = 200 / 0.20 x = 1000 Now that we know x is 1000, we can find 120% of x by multiplying x by 1.20 (since 120% is the same as 1.20 in decimal form): 1.20 * x = 1.20 * 1000 1.20 * x = 1200 So, 120% of the number is boxed{1200} .