# Solve the system of equations by the method of substitution.x^2 - y^2 = 9 x - y = 1

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We have to solve : x^2 - y^2 = 9 and x - y = 1 using the method of substitution.

Now x - y = 1

=> x = 1 + y

Substitute x = 1+ y in x^2 - y^2 = 9

=> ( 1 + y)^2 - y^2 = 9

=> 1 + y^2 + 2y - y^2 =9

=> 1+ 2y = 9

=> 2y = 9 -1

=> 2y = 8

=> y = 8/2

=> y = 4

Therefore y = 4. As x = 1 + y, x = 4+1 = 5.

**The required values of x and y are 5 and 4 respectively.**

We'll re-write the first equation:

x^2 - y^2 = 9 (1)

x - y = 1 (2)

To solve using the substitution method, we'll write x with respect to y, from the 2nd equation:

x = 1 + y (3)

We'll substitute (3) in (1):

(1+y)^2 - y^2 = 9 (4)

We notice that the equation (4) is a difference of squares:

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

We'll eliminate and combine like terms inside the brackets:

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

We'll re-write (4):

1 + 2y = 9

2y = 9 - 1

2y = 8

**y = 4**

We'll substitute y in (3):

x = 1 + 4

**x = 5**

**The solution of the system is {5 ; 4}.**

x^2-y^2 = 9......(1).

x-y = 1............(2).

From the second equation , we get -y = 1-x, or y = (x-1).

Substitute y = x-1 in equation (1) :

x^2- (x-1)^2 = 9.

x^2- (x^2-2x-1) = 9.

x^2-x^2+2x-1 = 9.

2x-1 = 9.

2x = 9+1 = 10.

x = 10/2 = 5.

Substitute x - 5 in eq (2): 5-y = 1. So 5 - 1 = y. So y = 4.

Therefore x = 5 and y = 4.