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1.3.6 Systems with Three or More Variables

Extend solving techniques to systems with three or more linear equations and variables.

定义

A system with three or more variables is a set of two or more linear equations involving three or more unknowns. A general system of \(n\) linear equations with \(n\) variables can be written as:

$\(\begin{cases}\na_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = b_1 \\\na_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = b_2 \\ \vdots \\\na_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = b_m \end{cases}\)$ \nwhere \(a_{ij}\) are coefficients, \(x_i\) are variables, and \(b_i\) are constants. The solution is an ordered \(n\)-tuple \((x_1, x_2, \ldots, x_n)\) that satisfies all equations simultaneously. Systems can have a unique solution, infinitely many solutions (dependent system), or no solution (inconsistent system).

核心公式

  • \(\begin{cases} a_1x + b_1y + c_1z = d_1 \ a_2x + b_2y + c_2z = d_2 \ a_3x + b_3y + c_3z = d_3 \end{cases}\) (Standard form of 3-variable system)
  • \(\text{Augmented Matrix: } \left[\begin{array}{ccc|c} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \end{array}\right]\) (Gaussian elimination representation)
  • \(\text{Row operations: } R_i \leftrightarrow R_j, \quad kR_i \rightarrow R_i \text{ (where } k \neq 0\text{)}, \quad R_i + kR_j \rightarrow R_i\) (Elementary row operations)
  • \(\text{Reduced Row Echelon Form (RREF): Leading 1 in each row, zeros above and below each leading 1}\) (Goal of Gaussian elimination)
  • \(\text{Determinant method (Cramer's Rule): } x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}\) (where \(D \neq 0\) for unique solution)

易错点

  • ⚠️ Forgetting to perform the same row operation on all elements in a row (including the augmented column) during Gaussian elimination, leading to incorrect coefficients and wrong solutions.
  • ⚠️ Misidentifying dependent vs. inconsistent systems: A row of all zeros indicates a dependent system (infinitely many solutions), while a row like \([0 \ 0 \ 0 | c]\) where \(c \neq 0\) indicates an inconsistent system (no solution).
  • ⚠️ Incorrectly applying Cramer's Rule when the determinant equals zero—students often try to divide by zero or fail to recognize that the system has either no solution or infinitely many solutions.
  • ⚠️ Making arithmetic errors when computing determinants for larger matrices, especially when expanding along rows or columns, resulting in incorrect values for \(D\), \(D_x\), \(D_y\), or \(D_z\).