# Solve the simultaneous equations y^2=x^2-9 and y= x-1 .

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### 2 Answers

We have to solve the equations:

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

y = x - 1 ...(2)

From (2) we get

y = x - 1

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

substitute in (1)

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

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

=> -2x = -10

=> x = 5

y = x - 1

=> y = 4

**The required solution is x = 5 and y = 4**

We'll subtract y^2 both sides, in the 1st equation:

x^2 -y^2 - 9 = 0

We'll add 9 both sides:

x^2 - y^2 = 9

We'll write the first equation as a difference of squares:

x^2 - y^2 = 9

(x - y)(x + y) = 9

We'll re-write the second equation:

x - y = 1

We'll substitute the second equation into the first:

1*(x+y)=9

x + y = 9

We'll change the second equation and we'll write y with respect to x.

y = x - 1

But x + y = 9

x + x - 1 = 9

We'll combine like terms:

2x - 1 = 9

2x = 10

x = 5

y = 5 - 1

y = 4

**The solution of the system is represented by the pair: {5 ; 4}.**