Given Information:
- a - b = 7
- a^2 - b^2 = 85

To Find:
a^3 - b^3

Solution:

Step 1: Understanding the Formula
To find a^3 - b^3, we can use the formula for the difference of cubes:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Step 2: Finding a + b
We are given that a - b = 7. By rearranging the equation, we can find a + b:

a - b = 7
a = b + 7

Step 3: Substituting a + b in the Formula
Now that we have found a + b, we can substitute it in the formula:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)
= (b + 7)(a^2 + ab + b^2)

Step 4: Finding a^2 + ab + b^2
We are given that a^2 - b^2 = 85. By rearranging the equation, we can find a^2 + ab + b^2:

a^2 - b^2 = 85
(a + b)(a - b) = 85
(a + b)(7) = 85
a + b = 85/7

Step 5: Substituting a^2 + ab + b^2 in the Formula
Now that we have found a^2 + ab + b^2, we can substitute it in the formula:

a^3 - b^3 = (b + 7)(a^2 + ab + b^2)
= (b + 7)(a + b)

Step 6: Finding the Value of a^3 - b^3
We have obtained the expression for a^3 - b^3 as (b + 7)(a + b). Now let's substitute the values of a + b and a - b:

a + b = 85/7
a - b = 7

We can solve these two equations simultaneously to find the values of a and b. Once we have the values of a and b, we can substitute them in the expression (b + 7)(a + b) to find the value of a^3 - b^3.

Conclusion:
To find the value of a^3 - b^3, we need to solve the given equations to obtain the values of a and b. Once we have the values of a and b, we can substitute them in the expression (b + 7)(a + b) to find the value of a^3 - b^3.