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