To solve the system of equations, we have two equations:
1. x² + y² = 25
2. y² = x² + 7
We can start by substituting the second equation into the first equation. From the second equation, we can express y² in terms of x:
y² = x² + 7
Now, we can substitute this expression for y² into the first equation:
x² + (x² + 7) = 25
This simplifies to:
2x² + 7 = 25
Next, we subtract 7 from both sides:
2x² = 25 – 7
2x² = 18
Now, divide both sides by 2:
x² = 9
Taking the square root of both sides gives us:
x = ±3
Now that we have the values for x, we can substitute these back into the equation for y²:
Using x = 3:
y² = 3² + 7 = 9 + 7 = 16
Thus, y = ±4.
Using x = -3:
y² = (-3)² + 7 = 9 + 7 = 16
Again, y = ±4.
Finally, the solution set of the system is:
(3, 4), (3, -4), (-3, 4), (-3, -4).
In summary, we found the possible pairs (x, y) that satisfy both equations in the given system.