To solve the system of equations:
- ( x + y = -1 )
- ( x^2 + y^2 = 25 )
We can use the first equation to express ( y ) in terms of ( x ):
From equation (1):
[
y = -1 - x
]
Next, substitute this expression for ( y ) into equation (2):
[
x^2 + (-1 - x)^2 = 25
]
Expanding the squared term:
[
x^2 + (1 + 2x + x^2) = 25
]
[
2x^2 + 2x + 1 = 25
]
Now, simplify this equation:
[
2x^2 + 2x + 1 - 25 = 0
]
[
2x^2 + 2x - 24 = 0
]
Divide the entire equation by 2:
[
x^2 + x - 12 = 0
]
Next, we can factor this quadratic equation:
[
(x + 4)(x - 3) = 0
]
Setting each factor to zero gives us:
[
x + 4 = 0 \quad \Rightarrow \quad x = -4
]
[
x - 3 = 0 \quad \Rightarrow \quad x = 3
]
Now we will find the corresponding values of ( y ) for each ( x ).
If ( x = -4 ):
[
y = -1 - (-4) = -1 + 4 = 3
]
Thus, one solution is ( (x, y) = (-4, 3) ).
If ( x = 3 ):
[
y = -1 - 3 = -4
]
Thus, the other solution is ( (x, y) = (3, -4) ).
The two solutions to the system of equations are:
[
(x, y) = (-4, 3) \quad \text{and} \quad (x, y) = (3, -4)
]
