If the original coordinate axes are rotated 45° to obtain the x' and y' axes, what is the value of x in terms of x' and y'?

as you know, the rotation matrix is

x' = x cosθ + y sinθ
y' = -x sinθ + y cosθ

You want the inverse matrix, which is the same as rotating (x',y') through -θ

x = x' cosθ - y' sinθ
y = x' sinθ + y' cosθ

Now just plug in θ = pi/4