[KaibaMath]1032 基于换元法解一道一元一次方程

解方程：(x-2)/2025 + (x-1)/2026 + (3-x)/1012 = 0.  

1049

 

下面给出相应的LaTex.

  1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  2 %               A Simple Proof Template
  3 %             Compiler: pdfLaTeX or XeLaTeX
  4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  5 
  6 \documentclass[11pt, a4paper]{amsart}
  7 
  8 % ===================================================================
  9 % PREAMBLE - PACKAGES AND SETUP
 10 % ===================================================================
 11 
 12 % --- Core Math Packages (amsart loads some, but we add others) ---
 13 \usepackage{amsmath}
 14 \usepackage{amssymb}
 15 
 16 % --- Visual Enhancement and Cross-Referencing ---
 17 \usepackage{tcolorbox}
 18 \usepackage{hyperref}
 19 \usepackage{cleveref}
 20 
 21 % --- Load tcolorbox libraries for theorem environments ---
 22 \tcbuselibrary{theorems, skins, breakable}
 23 
 24 % --- Hyperref setup for clean links ---
 25 \hypersetup{
 26     colorlinks=true,
 27     linkcolor=blue,
 28     filecolor=magenta,      
 29     urlcolor=cyan,
 30     citecolor=red,
 31 }
 32 
 33 % ===================================================================
 34 % CUSTOM ENVIRONMENTS AND COMMANDS
 35 % ===================================================================
 36 
 37 % --- Define a custom "Example" environment for problems ---
 38 \newtcbtheorem[number within=section]{example}{Problem}%
 39 {colback=cyan!5!white,colframe=pink!75!black,fonttitle=\bfseries}{ex}
 40 
 41 % --- Custom Math Commands ---
 42 \newcommand{\R}{\mathbb{R}}
 43 \newcommand{\N}{\mathbb{N}}
 44 \newcommand{\Z}{\mathbb{Z}}
 45 \newcommand{\Q}{\mathbb{Q}}
 46 
 47 % ===================================================================
 48 % DOCUMENT INFORMATION
 49 % ===================================================================
 50 
 51 \title{Solving a Linear Equation by Substitution}
 52 \author{Cyrus Li}
 53 \date{December 7, 2025}
 54 
 55 % ===================================================================
 56 % DOCUMENT BODY
 57 % ===================================================================
 58 
 59 \begin{document}
 60 
 61 \maketitle
 62 
 63 \section{Solving a Linear Equation}
 64 
 65 \begin{example}{Solve the following equation}{ex:rational_eq}
 66     \[ \frac{x-2}{2025} + \frac{x-1}{2026} + \frac{3-x}{1012} = 0 \]
 67 \end{example}
 68 
 69 \subsection{Solution}
 70 
 71 Set \( \alpha = 2025 \) and let \( y = x - 2 \). Then \( x = y + 2 \).
 72 
 73 \textbf{The original equation can be rewritten as:}
 74 \[ \frac{y}{\alpha} + \frac{y+1}{\alpha + 1} + \frac{2(1-y)}{\alpha - 1} = 0 \]
 75 
 76 \textbf{Multiplying both sides by the common denominator }
 77 \[\alpha(\alpha+1)(\alpha-1)\] 
 78 \textbf{ gives:}
 79 \begin{equation*}
 80 \begin{split}
 81 & y(\alpha+1)(\alpha-1) + (y+1)\alpha(\alpha-1) \\
 82 & \qquad + 2(1-y)\alpha(\alpha+1) = 0
 83 \end{split}
 84 \end{equation*}
 85 
 86 \textbf{Expanding and combining like terms:}
 87 \begin{equation*}
 88 \begin{split}
 89 & y(\alpha^2-1) + y(\alpha^2-\alpha) + (\alpha^2-\alpha) \\
 90 & \qquad + 2\alpha(\alpha+1) - 2y\alpha(\alpha+1) = 0
 91 \end{split}
 92 \end{equation*}
 93 \[ y\left[ (\alpha^2-1) + (\alpha^2-\alpha) - 2(\alpha^2+\alpha) \right] + \left[ (\alpha^2-\alpha) + 2(\alpha^2+\alpha) \right] = 0 \]
 94 \[ y(-1 - 3\alpha) + (3\alpha^2+\alpha) = 0 \]
 95 \[ y(3\alpha+1) = 3\alpha^2+\alpha \]
 96 \[ y(3\alpha+1) = \alpha(3\alpha+1) \]
 97 \[ y = \alpha \]
 98 
 99 \textbf{Therefore, we find } \( y = \alpha = 2025 \).
100 
101 \textbf{Thus, } \( x = y + 2 \text{, that is:} \)
102 \[ x = 2025 + 2 \]
103 \[ x = 2027 \] \qed
104 
105 \subsection{Conclusion}
106 A quick check confirms that \(x = 2027\) is a solution to the original equation.
107 
108 \end{document}

通过Overleaf编译成pdf截图如下：

 LaTeXP1049

 

posted @ 2025-12-07 10:38  CyrusLi  阅读(7)  评论(0)    收藏  举报