Find all solutions to the following system of equations (x-2)²+(y+3)²=4 ; x-y=3. illustrate with a graph.

The solutions are the points (0, -3) and (2, -1)

Step-by-step 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.

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step-by-step explanation:

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