The line y - x = 1 can be rewritten as y = x + 1.
The curve x = y^2 can be rewritten as y = sqrt(x).

To find the shortest distance between the line and the curve, we need to find the point on the curve that is closest to the line.
We can do this by finding the point of intersection between the line and the curve.

Substituting y = x + 1 into the equation x = y^2, we get:
x = (x + 1)^2
Expanding and rearranging, we get:
x = x^2 + 2x + 1
Rearranging again, we get:
x^2 + x - 1 = 0

Solving this quadratic equation, we get:
x = (-1 ± sqrt(1 + 4))/2
x = (-1 ± sqrt(5))/2

Since y = x + 1, we can substitute these values of x into y = x + 1 to find the corresponding values of y.

For x = (-1 + sqrt(5))/2:
y = (-1 + sqrt(5))/2 + 1
y = (-1 + sqrt(5))/2 + 2/2
y = (1 + sqrt(5))/2

For x = (-1 - sqrt(5))/2:
y = (-1 - sqrt(5))/2 + 1
y = (-1 - sqrt(5))/2 + 2/2
y = (1 - sqrt(5))/2

So, the two points of intersection between the line and the curve are:
((-1 + sqrt(5))/2, (1 + sqrt(5))/2) and
((-1 - sqrt(5))/2, (1 - sqrt(5))/2)

To find the shortest distance between the line and the curve, we need to find the distance between these two points.

Using the distance formula, the distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Plugging in the values for the two points of intersection, we get:
d = sqrt(((-1 - sqrt(5))/2 - (-1 + sqrt(5))/2)^2 + ((1 - sqrt(5))/2 - (1 + sqrt(5))/2)^2)
d = sqrt((2sqrt(5))^2 + (2sqrt(5))^2)
d = sqrt(4(5 + 5))
d = sqrt(40)
d = 2sqrt(10)

Therefore, the shortest distance between the line y - x = 1 and the curve x = y^2 is 2sqrt(10), which is approximately 6.325.