Solve the system (x (x - 1)^2 -(y - 2)^2 = (x - y)(x + y)

First, to calculate the addition of the 2 ratios from the
first equation, we'll calculate the LCD.

LCD = 5*2 =
10

2*(x - 2)/5 +5y/2 =
6*10

We'll re-write the first
equation:

2(x-2) + 5y =
60

We'll remove the
brackets:

2x - 4 + 5y =
60

We'll add 4 both sides:

2x
+ 5y = 60 + 4

2x + 5y = 64
(3)

Now, we'll expande the squares from the second equation
of the system:

(x - 1)^2 -(y - 2)^2 = (x - y)(x + y) -
59

x^2 - 2x + 1 - y^2 + 2y - 1 = (x - y)(x + y) -
59

We'll remove the brackets from the right
side:

(x - y)(x + y) - 59 = x^2 + xy - xy - y^2 -
59

x^2 - 2x + 1 - y^2 + 2y - 1 = x^2 + xy - xy - y^2 -
59

We'll eliminate like
terms:

- 2x + 2y = - 59
(4)

We'll add (3) to (4) and we'll
get:

2x + 5y - 2x + 2y = 64 -
59

We'll eliminate like
terms:

7y = 5

We'll divide by
7:

y =
5/7

We'll substitute y in
(3):

2x + 5*5/7 = 64

2x + 25/7
= 64

7*2x + 25 = 7*64

14x + 25
= 448

We'll subtract 25 both
sides:

14x = 448-25

14x =
423

x =
423/14

The solution of the
system is: {423/14 ; 5/7}.