Answer:
The solutions are the points (0, -3) and (2, -1)
Explanation:
Hi!
Let´s write the system of equations:
(x-2)² + (y + 3)² = 4
x - y = 3
The solutions of the systems are points (x,y) that satisfy both equations.
So, let´s take the second equation and solve it for x:
x - y = 3
Subtract x at both sides of the equation
-y = 3 - x
divide both sides for -1
y = x -3
Now, with this expression, we can replace y in the first equation:
(x-2)² + (y + 3)² = 4
(x-2)² + (x-3 +3)² = 4
(x-2)·(x-2) + x² = 4
Apply distributive property
x² - 4x + 4 + x² = 4
subtract 4 from both sides of the equation
2x² -4 x = 0
2x(x - 2) = 0
2x = 0 ⇒ x = 0
x-2 = 0
Add 2 to both sides
x = 2
Now, let´s calculate the values of y for x = 0 and x = 2
y = x -3
x = 0
y = 0 - 3 ⇒ y = -3
x = 2
y = 2 - 3 ⇒ y = -1
The solutions are the points (0, -3) and (2, -1)
Please see the attached figure. The points where the curve intersect are the solutions of the system.