1.1.2 Solving Linear Equations Using Inverse Operations

Apply inverse operations (addition, subtraction, multiplication, division) to isolate the variable and solve for x.

定义

Solving linear equations using inverse operations is a fundamental algebraic technique that involves isolating the variable (typically \(x\)) on one side of an equation by applying inverse operations systematically. An inverse operation is an operation that reverses the effect of another operation: addition is the inverse of subtraction, multiplication is the inverse of division, and vice versa. For a linear equation of the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\), we apply inverse operations in reverse order of operations (PEMDAS/BODMAS) to solve for the variable. The goal is to transform the equation into the form \(x = \text{value}\) by maintaining equality on both sides—whatever operation is performed on one side must be performed on the other side.

核心公式

• \(ax + b = c \Rightarrow ax = c - b \Rightarrow x = \frac{c - b}{a}\)
• \(\text{If } x + a = b, \text{ then } x = b - a\)
• \(\text{If } x - a = b, \text{ then } x = b + a\)
• \(\text{If } ax = b, \text{ then } x = \frac{b}{a}\) (where \(a \neq 0\))
• \(\text{If } \frac{x}{a} = b, \text{ then } x = ab\)

易错点

• ⚠️ Forgetting to apply the same operation to both sides of the equation, which violates the principle of equality and leads to incorrect solutions
• ⚠️ Incorrectly applying inverse operations in the wrong order—for example, dividing before subtracting when the correct order requires subtracting first (reverse of PEMDAS)
• ⚠️ Making sign errors when moving terms across the equals sign, such as changing \(x + 5 = 12\) to \(x = 12 + 5\) instead of \(x = 12 - 5\)
• ⚠️ Dividing by zero or failing to check that the coefficient of the variable is non-zero before dividing, which can occur in more complex equations