HL congruence for triangles occurs when x = -2.5 and y = 2.5.
For the triangles in the image to be congruent by HL, the hypotenuse and one leg of each triangle must be equal. The hypotenuse of the first triangle is 3y + x, and the hypotenuse of the second triangle is x + 5. The legs of the first triangle are y - x and x + 5, and the legs of the second triangle are y + 5 and x. Therefore, we have the following system of equations:
3y + x = x + 5
y - x = y + 5
Solving the first equation for x, we get:
x = 5 - 3y
Substituting this into the second equation, we get:
y - (5 - 3y) = y + 5
Simplifying, we get:
4y = 10
Therefore, y = 2.5. Substituting this value of y back into the equation for x, we get:
x = 5 - 3 * 2.5
Therefore, x = -2.5.
Therefore, the values of x and y that make the triangles congruent by HL are x = -2.5 and y = 2.5.