# Solve the system (x - 2)/5 +y/2 = 6 (x - 1)^2 -(y - 2)^2 = (x - y)(x + y) - 59

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(x - 2)/5 +y/2 = 6.............(1)

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

First let us re-write both equations:

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

Let us multiply by 10:

==> 2(x-2) + 5y = 60

==> 2x + 5y = 64 ........(3)

Now for equation (2):

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

==> x^2 - 2x + 1 - y^2 + 4y - 4 = x^2 - y^2 - 59

==> -2x + 4y - 3 = -59

==> -2x + 4y = -56 .........(4)

Now let us add (3) and (4):

==> 9y = 8

**==> y = 8/9**

Now to calculate x, we will substitute in (4):

2x = 4y + 56

==> x = 2y + 28

==> x= 2*8/9 + 28 = 16/9 + 28 = 268/9

**==> x= 268/9**

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}.**

To solve the equations:

(x - 2)/5 +y/2 = 6.........(1)

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

Multiply the equation (1) by 10 :

2(x-2) +5y = 60.

2x-4 +5y = 60

2x + 5y = 64.......(3)

Now simplify eq (2):(x - 1)^2 -(y - 2)^2 = (x - y)(x + y) - 59

x^2-2x+1- y^2+4y-4 = x^2-y^2-59

-2x+1+4y-4 = -59 , as x^2 and y^2 terms gets cancelled.

-2x+4y = -56 Divide by -1

2x-4y = 56....(4) .

2x+5y = 64....(3)

Therefore eq(3) - eq(4) gives: 9y = 8. So y = 8/9

From (3) : 2x+5y = 64. Or 2x+5*(8/9) = 64. So 2x = 64-40/9 = 238/9

So x = 268/9.

Therfore x = 268/9 and y = 8/9

The given system of equations to solve for x and y is:

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

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

We go about it as follows:

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

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

=> 2x - 4 + 5y = 60

=> 2x + 5y = 64

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

=> x^2 +1 - 2x - y^2 - 4 + 4y = x^2 - y^2 - 59

=> 1 -2x - 4 +4y = -59

=> -2x +4y= -56

=> x- 2y = 28

Now using

2x + 5y = 64 ...(1)

x- 2y = 28....(2)

(1)- 2*(2)

=> 2x + 5y - 2x + 4y = 64 - 56

=> 9y = 8

=> y= 8/9

Now substitute y = 8/9 in (2)

x- 2(8/9) = 28

=> x = 28 + 16/9

=> x = 268/9

**Therefore x= 268/9 and y = 8/9**