To solve the system of equations:
- ( x^2 + y^2 = 61 )
- ( x - y + 10 = 9 )
Let's start with the second equation to express (y) in terms of (x):
From the equation ( x - y + 10 = 9 ):
[
x - y = -1
]
This simplifies to:
[
y = x + 1
]
Now, we can substitute (y = x + 1) into the first equation:
[
x^2 + (x + 1)^2 = 61
]
Expanding the second term:
[
x^2 + (x^2 + 2x + 1) = 61
]
Combining like terms:
[
2x^2 + 2x + 1 = 61
]
Next, we subtract 61 from both sides:
[
2x^2 + 2x + 1 - 61 = 0
]
This simplifies to:
[
2x^2 + 2x - 60 = 0
]
We can divide the entire equation by 2:
[
x^2 + x - 30 = 0
]
Now, we can factor this quadratic equation:
[
(x + 6)(x - 5) = 0
]
Setting each factor equal to zero gives:
[
x + 6 = 0 \quad \Rightarrow \quad x = -6
]
[
x - 5 = 0 \quad \Rightarrow \quad x = 5
]
Now, we can find the corresponding (y) values for each (x):
For (x = -6):
[
y = -6 + 1 = -5
]
For (x = 5):
[
y = 5 + 1 = 6
]
Thus, the solution pairs ((x, y)) are:
[
(-6, -5) \quad \text{and} \quad (5, 6)
]
So the final solutions to the system of equations are:
[
(x, y) = (-6, -5) \quad \text{and} \quad (5, 6)
]
