answer:Since alpha is in the interval (pi, 2pi), and the cosine of alpha is given by cosalpha = -frac{sqrt{5}}{5}, we can determine the sine of alpha using the Pythagorean identity: sin^2alpha + cos^2alpha = 1 Solving this for sinalpha, keeping in mind that sinalpha will be negative in the given interval because alpha is in the third or fourth quadrant, we find: sinalpha = -sqrt{1 - cos^2alpha} = -sqrt{1 - left(-frac{sqrt{5}}{5}right)^2} = -sqrt{1 - frac{5}{25}} = -sqrt{frac{20}{25}} = -frac{2sqrt{5}}{5} Next, we calculate tanalpha: tanalpha = frac{sinalpha}{cosalpha} = frac{-frac{2sqrt{5}}{5}}{-frac{sqrt{5}}{5}} = 2 Finally, we use the double angle identity for tangent to find tan2alpha: tan2alpha = frac{2tanalpha}{1 - tan^2alpha} = frac{2 cdot 2}{1 - (2)^2} = frac{4}{1 - 4} = -frac{4}{3} Thus, the value of tan2alpha is -frac{4}{3}. boxed{tan2alpha = -frac{4}{3}}

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answer:To explore the relationship between x, y, and z, we can start by solving the first equation for x, giving x = yz/2. Now, substituting this expression for x into the second equation, we get 5y = (yz/2)z. Simplifying, we find 5y = yz^2/2, and rearranging terms yields 10y = yz^2. Assuming y is non-zero (since if y = 0, the equation 5y = xz implies xz = 0, leading to either x = 0, z = 0, or both), we can divide both sides by y to obtain z^2 = 10. This reveals that z can only be one of two possible values: the square root of 10 or the negative square root of 10. Knowing z, we can then express x and y in terms of z, but their precise values will depend on the chosen value of z. However, regardless of the specific value of z, we can deduce that x, y, and z are all related through z, with x and y being proportional to each other. In essence, once z is fixed, the values of x and y become dependent on each other through their shared relationship with z.

answer:Now that we know z can only be the square root of 10 or the negative square root of 10, we can derive the relationships between x, y, and z. Recall the equation 10y = yz^2, which we used to find z. We also have the equation 2x = yz. Since z^2 = 10, we can substitute z^2 into the original equation 5y = xz to get 5y = x sqrt(10) or 5y = x(-sqrt(10)) depending on the chosen value of z. If we divide the equation 2x = yz by the equation 5y = xz, we get (2x)/(5y) = yz / xz, which simplifies to 2/5 = y/x. Therefore, we can say that y is proportional to x, with y = 2x/5. Given the value of z, we can express x and y in terms of each other. For instance, using the equation 2x = yz and substituting z = sqrt(10) or z = -sqrt(10), we can solve for x or y in terms of the other variable and z. The specific value of x will depend on the chosen value of y, and vice versa. These relationships highlight the intimate connection between x, y, and z, with z acting as a parameter that constrains the values of x and y.