Systems of Nonlinear Equations and Inequalities: Learn It 3

Solving a System of Nonlinear Equations Using Elimination

We have seen that substitution is often the preferred method when a system of equations includes a linear equation and a nonlinear equation. However, when both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps.

It is not necessary that you know the form of an equation given in a system in order to solve the system (such as with an ellipse or hyperbola in the examples below) . If you are unable to sketch the graph of an equation given in the system, you must be extra diligent with your algebra though to avoid missing or extraneous solutions.

ellipse

An ellipse is defined as the set of all points where the sum of the distances from two fixed points (the foci) is a constant. It is an “elongated” or “squashed” circle, characterized by a center, a major axis (the longest diameter) and a minor axis (the shortest diameter) perpendicular to the major axis. A circle is a special case of an ellipse where the two foci are at the same location

possible types of solutions for the points of intersection of a circle and an ellipse

The figure below illustrates possible solution sets for a system of equations involving a circle and an ellipse.

  • No solution. The circle and ellipse do not intersect. One shape is inside the other or the circle and the ellipse are a distance away from the other.
  • One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.
  • Two solutions. The circle and the ellipse intersect at two points.
  • Three solutions. The circle and the ellipse intersect at three points.
  • Four solutions. The circle and the ellipse intersect at four points.

This image illustrates the different possible intersections between a circle and an ellipse. In the first case, there is no intersection, resulting in no solutions. In the second case, the circle and ellipse touch at one point, indicating one solution. In the third case, the shapes intersect at two points, resulting in two solutions. In the fourth case, they intersect at three points, showing three solutions. In the fifth case, the circle and ellipse intersect at four points, yielding four solutions.

Solve the system of nonlinear equations.

[latex]\begin{align} {x}^{2}+{y}^{2}=26 \hspace{5mm} \left(1\right)\\ 3{x}^{2}+25{y}^{2}=100 \hspace{5mm} \left(2\right)\end{align}[/latex]