1.1.4 Equations with Variables on Both Sides

Solve linear equations where the variable appears on both sides by collecting variables on one side.

定义

An equation with variables on both sides is a linear equation where the variable (typically \(x\)) appears in expressions on both the left and right sides of the equal sign. To solve such equations, the fundamental strategy is to collect all variable terms on one side of the equation and all constant terms on the other side, using inverse operations (addition/subtraction and multiplication/division) to isolate the variable. The general form can be written as \(ax + b = cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants and \(a \neq c\). The solution process involves: (1) subtracting or adding terms to move all variables to one side, (2) simplifying to get \((a-c)x = d-b\), and (3) dividing by the coefficient to find \(x = \frac{d-b}{a-c}\).

核心公式

• \(ax + b = cx + d\)
• \((a-c)x = d - b\)
• \(x = \frac{d-b}{a-c}\) (where \(a \neq c\))
• \(ax + b = cx + d \Rightarrow ax - cx = d - b \Rightarrow (a-c)x = d-b\)
• \(2x + 5 = x + 12 \Rightarrow 2x - x = 12 - 5 \Rightarrow x = 7\)

易错点

• ⚠️ Forgetting to apply the same operation to both sides of the equation when moving terms, leading to incorrect solutions
• ⚠️ Making sign errors when moving terms across the equal sign (e.g., changing \(+5\) to \(-5\) or vice versa incorrectly)
• ⚠️ Failing to combine like terms properly before isolating the variable, resulting in algebraic errors
• ⚠️ Dividing by zero or forgetting to check that the coefficient of the variable is non-zero after collecting variable terms