To find the values of x and y, we can use the system of equations given:

1) x^2 + y^2 = 19
2) x*y = 3

First, we can rearrange equation 2) to solve for y in terms of x:

y = 3 / x

Next, we can substitute this into equation 1):

x^2 + (3/x)^2 = 19
x^2 + 9/x^2 = 19

Multiplying through by x^2:

x^4 + 9 = 19x^2
x^4 - 19x^2 + 9 = 0

Let z = x^2:

z^2 - 19z + 9 = 0

Using the quadratic formula to solve for z:

z = [19 ± √(19^2 - 419)] / 2
z = [19 ± √(361 - 36)] / 2
z = [19 ± √325] / 2

z1 = (19 + √325) / 2
z2 = (19 - √325) / 2

Since z = x^2, we take the square root of z to find x:

x1 = √[(19 + √325) / 2]
x2 = √[(19 - √325) / 2]

From the equation y = 3 / x, we can find the corresponding values of y for x1 and x2:

y1 = 3 / x1
y2 = 3 / x2

Calculating these values will give us the solutions for x and y.