If x² + y² =1, and the average of x + y =4, what is x – y?

Considering that this question is asking for "x - y" rather than their individual values, it is highly implied that there might be shortcuts depending on how you algebraically manipulate things.

However, the equation x² + y² = 1 is an equation of a unit circle (circle of radius 1 at origin). Additionally x + y = 4 is a straight line with intercepts at (0, 8) & (8, 0). This means that these two equations never intersect hence no solution for x and y that satisfy both equations. Something seems wrong unless you want imaginary solutions.

Anyhow, this is the standard method I have been using:

average(x + y) = 4

(x + y) / 2 = 4

x + y = 8

y = 8 - x

Now substituting this into another equation:

x² + (8 - x)² = 1

x² + (64 + x² - 16x) = 1

2x² - 16x + 63 = 0

Now solving for imaginary roots:

x² - 8x + 63/2 = 0

(x - 4)² - 16 + 31.5 = 0

x = +4 ± √(-15.5) = 4 ± 3.94i

Therefore, y = 8 - x = 8 - (4 ± 3.94i) = +4 (- or +) 3.94i

So the pair of solutions are as follows: (4 + 3.94i, 4 - 3.94i) , (4 - 3.94i, 4 + 3.94i)

So now,

x - y = [ (4 + 3.94i) - (4 -3.94i) ] = 7.87i

OR

x - y = [ (4 - 3.94i) - (4 + 3.94i) ] = -7.87i