To solve for the values of x and y in the equations:

X + y = 11 ............(1) xy = 24 ............(2)

First, we can solve for one variable in terms of the other. From equation (1), we can isolate y by subtracting x from both sides:

y = 11 - x

Then, we substitute this expression for y into equation (2) to obtain an equation in terms of x:

x(11 - x) = 24

Expanding the left-hand side:

11x - x^2 = 24

Rearranging terms:

x^2 - 11x + 24 = 0

This is a quadratic equation that we can solve using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 1, b = -11, and c = 24. Substituting these values:

x = (-(-11) ± sqrt((-11)^2 - 4(1)(24))) / 2(1) x = (11 ± sqrt(121 - 96)) / 2 x = (11 ± sqrt(25)) / 2

Taking the positive square root:

x = (11 + 5) / 2 = 8 or x = (11 - 5) / 2 = 3

Now that we have the values of x, we can use equation (1) to solve for y. If x = 8:

8 + y = 11 y = 3

If x = 3:

3 + y = 11 y = 8

Therefore, the solutions to the equations are x = 8, y = 3 and x = 3, y = 8.