Okay, let's try to figure out how to find the value of y here. The problem says that the projection of the vector (2, y, -5) onto the vector (1, -2, 1) is equal to (5/6) times (1, -2, 1). So, I need to use the formula for the projection of one vector onto another and set it equal to that given projection, then solve for y. Let me recall the formula for the projection of a vector a onto vector b. I think it's (a · b / ||b||²) * b. Right? So, the projection of a onto b is a scalar multiple of b, where the scalar is the dot product of a and b divided by the magnitude squared of b. 

Given that, let's denote the vectors. Let’s call vector a as (2, y, -5) and vector b as (1, -2, 1). The projection of a onto b is (a · b / ||b||²) * b, which according to the problem is equal to (5/6) * (1, -2, 1). So, if I compute (a · b / ||b||²) * b, it should equal (5/6) * (1, -2, 1). Therefore, the scalar multiplier (a · b / ||b||²) must be equal to 5/6. So, first, let's compute the dot product of a and b, then divide by the magnitude squared of b, set that equal to 5/6, and solve for y.

Let me compute the dot product first. a · b is 2*1 + y*(-2) + (-5)*1. That simplifies to 2 - 2y -5. Which is (2 -5) -2y = -3 -2y. So, a · b = -3 -2y.

Now, compute ||b||². The magnitude squared of vector b is 1² + (-2)² + 1². That is 1 + 4 + 1 = 6. So, ||b||² is 6.

So, the scalar multiplier is (a · b) / ||b||² = (-3 -2y)/6. According to the problem, this scalar is equal to 5/6. Therefore, we can set up the equation:

(-3 -2y)/6 = 5/6

Multiply both sides by 6 to eliminate the denominator:

-3 -2y = 5

Then, solve for y:

-2y = 5 + 3

-2y = 8

Divide both sides by -2:

y = 8 / (-2) = -4

Wait, so y is -4? Let me check my steps again to make sure I didn't make a mistake.

First, the projection formula: (a · b / ||b||²) * b. So, the scalar is (a · b)/||b||². Then, we set that scalar equal to 5/6 because the projection is 5/6 times vector b. So, (a · b)/||b||² = 5/6. 

Calculating a · b: 2*1 + y*(-2) + (-5)*1 = 2 -2y -5 = -3 -2y. Correct.

||b||²: 1 + 4 + 1 = 6. Correct.

So, (-3 -2y)/6 = 5/6. Multiply both sides by 6: -3 -2y = 5. Then, -2y = 5 +3 = 8. So, y = -4. That seems right. Let me verify by plugging y = -4 back into the projection.

Compute a · b with y = -4: 2*1 + (-4)*(-2) + (-5)*1 = 2 +8 -5 = 5. Then, ||b||² is 6. So, 5/6. Therefore, the scalar is indeed 5/6, so the projection is 5/6*(1, -2, 1). Which matches the given projection. So, the answer y = -4 is correct. 

I think that's it. Let me just recap to make sure I didn't skip any steps or make any calculation errors. The key was to use the projection formula, set the scalar equal to 5/6, solve for y. The dot product calculation was crucial, and then solving the linear equation for y. Checked the result by plugging back in, which worked out. So, confident that y is -4.
Therefore, the value of $y$ is $\boxed{-4}$.
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