Find the solutions to the system of nonlinear equations given by: \[ \left\{\begin{array}{l} y=3 x+1 \\ y=-x^{2}+1 \end{array}\right. \] Enter your answer as a list of ordered pair. For example: $(2,-5),(5,-2)$

Step 1 ：The system of equations is nonlinear because one of the equations is quadratic. To find the solutions to the system, we need to find the values of x and y that satisfy both equations simultaneously.

Step 2 ：One way to do this is to set the two equations equal to each other and solve for x. Then, we can substitute the x-values into one of the original equations to find the corresponding y-values.

Step 3 ：Let's start by setting the two equations equal to each other: \(3x + 1 = -x^2 + 1\)

Step 4 ：This simplifies to: \(x^2 + 3x = 0\)

Step 5 ：We can solve this quadratic equation for x by factoring: \(x(x + 3) = 0\)

Step 6 ：Setting each factor equal to zero gives us the solutions x = 0 and x = -3.

Step 7 ：Next, we substitute these x-values into the first equation \(y = 3x + 1\) to find the corresponding y-values.

Step 8 ：For x = 0, y = 1. For x = -3, y = -8.

Step 9 ：Final Answer: The solutions to the system of nonlinear equations are \(\boxed{(0, 1), (-3, -8)}\).