Nonlinear systems Knowledge-Based Study

Study notes

1. What is a Nonlinear System?

   A system of nonlinear equations is a set of two or more equations involving two or more variables, where at least one equation is not linear. In a linear equation, all variables are raised to the first power and there are no products of variables. In a nonlinear equation, terms may contain exponents greater than one, square roots, products of variables, or other nonlinear functions.

   Geometrically, solving a system corresponds to finding the coordinates where the graphs of the equations intersect.

   • Linear systems: Can have zero, one, or infinitely many solutions.
   • Nonlinear systems: Can have zero, one, two, three, four, or more distinct real solutions, depending on the shapes and positions of the interacting curves.

   The Substitution Method

   The method of substitution is especially effective when one of the equations in the system is linear or can be easily solved for one variable in terms of the other.

   Step-by-Step Procedure

   1. Isolate one variable in the simpler (often linear) equation.
   2. Substitute the resulting expression into the nonlinear equation to create a single-variable equation.
   3. Solve the resulting equation for that variable.
   4. Substitute each solution back into the isolated expression to determine the corresponding values of the other variable.
   5. Write each solution as an ordered pair $(x, y)$ and verify each pair in the original system.

   Example

   Solve the system:

   $$\begin{cases} y = x^2 - 3x + 2 \\ y = x - 1 \end{cases}$$

   Substitute $y = x - 1$ into the quadratic equation:

   $$x - 1 = x^2 - 3x + 2$$

   Rearrange into standard quadratic form:

   $$x^2 - 4x + 3 = 0$$

   Factoring gives $(x - 1)(x - 3) = 0$, which yields $x = 1$ and $x = 3$.

   Find the corresponding $y$-values using $y = x - 1$:

   • For $x = 1$: $y = 1 - 1 = 0$ $\rightarrow$ Point: $(1, 0)$
   • For $x = 3$: $y = 3 - 1 = 2$ $\rightarrow$ Point: $(3, 2)$

   The solution set consists of the two points of intersection: $(1, 0)$ and $(3, 2)$.

   The Elimination Method

   The elimination (or addition) method is useful when both equations share like-degree terms, such as both having $x^2$ or $y^2$ terms that can be eliminated through linear combinations.

   Example

   Solve the system:

   $$\begin{cases} x^2 + y^2 = 25 \\ x^2 - y^2 = 7 \end{cases}$$

   Add the two equations directly to eliminate $y^2$:

   $$(x^2 + y^2) + (x^2 - y^2) = 25 + 7$$

   $$2x^2 = 32 \implies x^2 = 16 \implies x = \pm 4$$

   Substitute $x^2 = 16$ into the first equation to solve for $y$:

   $$16 + y^2 = 25 \implies y^2 = 9 \implies y = \pm 3$$

   Pairing each $x$-value with each possible $y$-value yields four distinct real solutions:

   $(4, 3)$, $(4, -3)$, $(-4, 3)$, and $(-4, -3)$.

   Geometrically, this represents the four intersection points between a circle centered at the origin and a hyperbola centered at the origin.

   Interpreting Solution Sets Geometrically

   The algebraic solutions of a system directly mirror the geometric relationship between the graphs:

   • Secant Line (Two Solutions): A line intersects a curve (such as a circle or parabola) at two distinct points. In this case, the resulting single-variable quadratic equation has a positive discriminant ($\Delta > 0$).
   • Tangent Line (One Solution): A line touches a curve at exactly one point without crossing through it. Here, the quadratic equation yields a repeated root ($\Delta = 0$).
   • Non-intersecting (No Real Solutions): The curves do not meet in the real plane. The resulting quadratic equation produces no real roots ($\Delta < 0$).

   Important Verification Note

   Always substitute potential solutions into both original equations. In cases involving square roots or rational expressions, algebraic operations may introduce extraneous solutions that do not satisfy the original system.