Lesson 5: Solving Nonlinear Systems

Property

A system of nonlinear equations is a system where at least one of the equations is not linear.

Just as with systems of linear equations, a solution of a nonlinear system is an ordered pair that makes both equations true. In a nonlinear system, there may be more than one solution. The graphs may be circles, parabolas or hyperbolas and there may be several points of intersection, and so several solutions.

Examples

• A system with a parabola and a circle:

Property

The solution to a system of equations is the set of intersection points of their graphs. For a system with a quadratic equation, we can solve it algebraically. If both equations are solved for $y$, we can set the expressions equal to each other. This creates a single equation in terms of $x$, which can then be solved. The resulting equation is often quadratic and can be solved using methods like the quadratic formula.

Examples

• To solve the system $y = x^2 - 2x + 3$ and $y = x + 1$, set the expressions for $y$ equal: $x^2 - 2x + 3 = x + 1$. This simplifies to $x^2 - 3x + 2 = 0$, which factors into $(x-1)(x-2)=0$. The solutions are $x=1$ and $x=2$. The corresponding points are $(1, 2)$ and $(2, 3)$.

• A company's cost is $C = 0.1x^2 + 2x + 100$ and revenue is $R = 12x$. To break even, $C=R$. So, $0.1x^2 + 2x + 100 = 12x$. This gives $0.1x^2 - 10x + 100 = 0$. Using the quadratic formula, break-even points are at $x=11.27$ and $x=88.73$ items.