The video begins by emphasizing the importance of checking all possible solutions for systems of nonlinear equations, as not all generated solutions may be correct. It demonstrates this by checking two solutions for a given system, showing that both satisfy the original equations.

The first detailed example involves two quadratic equations. The video uses the elimination method by multiplying one equation by -1 to enable cancellation of terms. After eliminating 'y' and 'x squared', the resulting linear equation is solved for 'x'. This 'x' value is then plugged back into one of the original equations to find 'y', and the solution is verified against the second original equation.

This example showcases the substitution method for a system with a quadratic and a linear equation. The linear equation is rearranged to solve for 'y', which is then substituted into the quadratic equation. The resulting quadratic equation in 'x' is simplified, factored, and solved for two possible 'x' values. These 'x' values are used to find corresponding 'y' values, yielding two solution pairs, which are then verified.

A third example demonstrates substitution for a system including a product term (x*y) and a linear equation. The linear equation is solved for 'y' and substituted into the first equation. This leads to a quadratic equation, which is simplified by dividing by a common factor. The quadratic is then factored to find two 'x' values, and subsequent 'y' values are determined, resulting in two valid solutions.

The final example uses elimination for two quadratic equations with squared terms. Adding the equations directly eliminates the 'y squared' term, allowing for the calculation of 'x squared'. This results in two possible values for 'x' (positive and negative). Plugging these back into one of the original equations (or a rearranged version of it) yields two possible values for 'y', leading to four distinct solution pairs. The video concludes by confirming that all four solutions satisfy both original equations.