Question

What are the solutions to the nonlinear systems of equations below x^2+y^2=4. x^2/2^2-y^2/2^2=1

Answer 1

Explanation:

To find the solutions to the nonlinear system of equations:

1. Start by solving the first equation x^2 + y^2 = 4. This equation represents a circle with a radius of 2 centered at the origin (0,0).

2. The second equation x^2/2^2 - y^2/2^2 = 1 can be simplified as x^2/4 - y^2/4 = 1. This equation represents a hyperbola centered at the origin (0,0) with the x-axis as its transverse axis.

3. To find the intersection points of the circle and hyperbola, we need to substitute one equation into the other.

4. Substitute x^2 + y^2 = 4 into x^2/4 - y^2/4 = 1:

(4-y^2)/4 - y^2/4 = 1

4 - y^2 - y^2 = 4

-2y^2 = 0

y^2 = 0

y = 0

5. Substituting y = 0 back into x^2 + y^2 = 4, we get x^2 + 0^2 = 4. This simplifies to x^2 = 4, which has two solutions: x = 2 and x = -2.

Therefore, the solutions to the nonlinear system of equations x^2 + y^2 = 4 and x^2/2^2 - y^2/2^2 = 1 are the two points (2, 0) and (-2, 0) where the circle and hyperbola intersect.

Answer 2

Explanation:

The nonlinear system of equations you provided consists of two equations:

1. x^2 + y^2 = 4

2. (x^2 / 2^2) - (y^2 / 2^2) = 1

To find solutions to this system, you can use substitution or elimination. Let's solve it using substitution.

From equation (2), you can rearrange it to isolate x^2:

(x^2 / 2^2) - (y^2 / 2^2) = 1

(x^2 / 4) - (y^2 / 4) = 1

x^2 - y^2 = 4

Now, you have a system of equations:

1. x^2 + y^2 = 4

2. x^2 - y^2 = 4

You can add equation (1) and equation (2) to eliminate y^2:

(x^2 + y^2) + (x^2 - y^2) = 4 + 4

2x^2 = 8

x^2 = 4

x = ±2

Now that you have found x, you can substitute it back into one of the original equations (e.g., equation (1)) to find the corresponding y-values:

For x = 2:

2^2 + y^2 = 4

4 + y^2 = 4

y^2 = 0

y = 0

For x = -2:

(-2)^2 + y^2 = 4

4 + y^2 = 4

y^2 = 0

y = 0

So, the solutions to the nonlinear system of equations are:

1. (x, y) = (2, 0)

2. (x, y) = (-2, 0)

These are the points where both equations are satisfied.