2.8.1 Systems of Nonlinear Equations¶

Solving systems containing two or more nonlinear equations simultaneously using substitution, elimination, and graphical methods.

定义¶

A system of nonlinear equations is a set of two or more equations where at least one equation is nonlinear (i.e., contains variables with exponents other than 1, products of variables, trigonometric functions, exponential functions, or other nonlinear operations). Formally, a system of nonlinear equations can be written as:

$$\begin{cases}\nf_1(x, y, \ldots) = 0 \\\nf_2(x, y, \ldots) = 0 \\ \vdots \\\nf_n(x, y, \ldots) = 0 \end{cases}$$ \nwhere at least one of the functions $f_i$ is nonlinear. Common examples include systems containing quadratic equations, circles, ellipses, hyperbolas, exponential equations, logarithmic equations, and trigonometric equations. Solutions to such systems are the ordered pairs (or tuples) that simultaneously satisfy all equations in the system. These solutions represent the intersection points of the curves defined by each equation.

核心公式¶

$x^2 + y^2 = r^2$ (circle equation)
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (ellipse equation)
$y = ax^2 + bx + c$ (parabola equation)
$xy = k$ (rectangular hyperbola equation)
$For a system with two equations, the number of real solutions equals the number of intersection points of the two curves$

易错点¶

⚠️ Forgetting to check all potential solutions by substituting back into both original equations, leading to extraneous solutions introduced during algebraic manipulation (especially when squaring both sides)
⚠️ Incorrectly assuming that a system has no real solutions without considering all possible solution methods or graphical interpretations; missing solutions that occur at points of tangency or multiple intersections
⚠️ Failing to recognize when substitution creates a higher-degree polynomial that may have more solutions than expected, or conversely, losing solutions when dividing by expressions that could equal zero
⚠️ Misinterpreting the geometric meaning of solutions; not understanding that the number and type of solutions depend on how the curves intersect (transversely, tangentially, or not at all)