Maths Puzzle 8 - Algebra

I was looking at sample questions for CBSE board for 9th grade (http://papers.icbse.com/sample/ix/2011/4.pdf).

Here is a problem and the solution.

Question No. 23:

Find the value of x³ + y³ + 15xy - 125 when x+y=5

There are two ways to solve this. Let's do it the formal way first

Solution 1:

Since x+y=5, y=5-x. Substitute this in the 1st equation

x³ + y³ + 15xy - 125 = x³ + (5-x)³ +15x(5-x) - 125

=x³ + (5³ - 3.5².x + 3.5.x² - x³) + (75x - 15x²) - 125

(using the formula for (a-b)³ )

=x³ + 125 - 75x + 15x² - x³ + 75x -15x² - 125 

= 0

Now let's go through an informal solution (note: this is just used to get a quick answer and it's not the "right" way...)

Solution 2:

Find the value of x³ + y³ + 15xy - 125 when x+y=5

Since there are two variables and x+y=5 with no other equation involving x and y, we may assume that it does not matter what the values of x and y are provided that x+y=5. So let's assume x=5, y=0

Substituting these values of x and y in the first equation we get,

x³ + y³ + 15xy - 125 = 5³ + 0 + 0 - 125 = 0

voila!!!