I have no clue how to do this as no values were given and im confused. can someone explain plz?
if x^3-y^3=m and x-y=N, express xy in terms of M and N
This is one of those pure manipulation problems.
you have to know that x^3-y^3 = (x-y)(x^2+xy+y^2)
so n(x^2+xy+y^2)=m
(x^2+xy+y^2) = m/n (#1)
notice we have xy, which is what we want, inside the bracket, but there is that x^2+y^2 hanging around
(from countless years of experience I know)
x^2+y^2 = (x-y)^2 + 2xy
so let's put that back in #1
((x-y)^2 + 2xy + xy) = m/n
n^2 + 3xy = m/n
3xy = m/n - n^2
3xy = (m-n^3)/n
and finally
xy = (m-n^3)/(3n)
I don't know what grade level you are in but that is pretty nasty.
To solve this problem, you need to understand some algebraic manipulations. Let's break it down step by step:
1. Start with the equation x^3 - y^3 = m. This is a special case known as a difference of cubes, which can be factored as (x - y)(x^2 + xy + y^2) = m.
2. Now, you are given that x - y = N. Substitute this value into the factored equation to get (N)(x^2 + xy + y^2) = m.
3. You want to express xy in terms of M and N. Notice that the equation contains xy along with x^2 and y^2 terms. To simplify it, you need to find an expression for x^2 + y^2 in terms of N.
4. Use the identity x^2 + y^2 = (x - y)^2 + 2xy. Plug in N for x - y to get x^2 + y^2 = N^2 + 2xy.
5. Substitute this expression back into the factored equation: (N)(N^2 + 2xy) = m.
6. Expand the equation to get N^3 + 2Nxy = m.
7. Now, solve for xy. Rearrange the equation to isolate the term with xy: 2Nxy = m - N^3.
8. Divide both sides of the equation by 2N to get xy = (m - N^3)/(2N).
So, the expression for xy in terms of M and N is xy = (m - N^3)/(2N).
It's important to note that this problem requires knowledge of factoring, algebraic identities, and manipulating equations. Remember to pay attention to the given information and try to identify patterns or identities that can help simplify the problem.