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1.1.3 Multi-Step Equation Solving

Solve linear equations requiring multiple operations, including combining like terms and distributing coefficients.

定义

Multi-step equation solving is the process of finding the value of an unknown variable in a linear equation that requires two or more operations to isolate the variable. This typically involves combining like terms, applying the distributive property, and using inverse operations (addition/subtraction and multiplication/division) in a systematic sequence. A multi-step linear equation has the general form \(ax + b = c\) or more complex variations such as \(a(x + b) + c = d\) or \(ax + b = cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants and \(x\) is the variable to be solved. The goal is to isolate the variable on one side of the equation to determine its value.

核心公式

  • \(ax + b = c \Rightarrow x = \frac{c - b}{a}\)
  • \(a(x + b) = c \Rightarrow x = \frac{c}{a} - b\)
  • \(ax + b = cx + d \Rightarrow (a - c)x = d - b \Rightarrow x = \frac{d - b}{a - c}\)
  • \(a(bx + c) + d = e \Rightarrow abx + ac + d = e \Rightarrow x = \frac{e - ac - d}{ab}\)
  • \(\frac{ax + b}{c} = d \Rightarrow ax + b = cd \Rightarrow x = \frac{cd - b}{a}\)

易错点

  • ⚠️ Incorrectly distributing coefficients: Students often forget to multiply all terms inside parentheses by the coefficient. For example, writing \(2(x + 3) = 2x + 3\) instead of \(2(x + 3) = 2x + 6\).
  • ⚠️ Failing to combine like terms before isolating the variable: Students may attempt to isolate the variable prematurely without first simplifying both sides of the equation, leading to incorrect intermediate steps.
  • ⚠️ Sign errors when moving terms across the equal sign: Students frequently forget to change the sign of a term when moving it to the opposite side of the equation. For instance, solving \(x + 5 = 12\) as \(x = 12 + 5\) instead of \(x = 12 - 5\).
  • ⚠️ Dividing only one term instead of the entire side by a coefficient: When solving \(2x + 4 = 10\), students might divide only the first term, writing \(x + 4 = 10\) instead of first subtracting 4 to get \(2x = 6\), then dividing both sides by 2.