Linear equations and inequalities

前置知识

9 项必需的前置知识（一直追溯到底）

你将学会

• Solve linear equations with the variable on both sides, including fractions and negative coefficients.
• Recognise equations with exactly one solution, no solution or infinitely many solutions.
• Solve linear inequalities, reversing the inequality when multiplying or dividing by a negative number.

定义

• A linear equation in x is one that can be written as ax + b = cx + d.
• A solution is a value of the variable that makes the equation true.
• Two equations are equivalent if they have the same solutions.

定理

• Adding the same number to both sides, or multiplying both sides by the same non-zero number, gives an equivalent equation.
• ax = b has exactly one solution if a ≠ 0, none if a = 0 and b ≠ 0, and every number as a solution if a = b = 0.
• Multiplying both sides of an inequality by a negative number reverses its direction.

常见误解

• Multiplying both sides by zero, or by an expression that might be zero, and expecting an equivalent equation.
• Forgetting to reverse an inequality after multiplying by a negative number.
• Applying an operation to only one term on a side of the equation.

通向

可选发现

• Linear algebra Systems of many linear equations in many unknowns are the starting point of linear algebra.

学习路径

• Adult beginner pathway

延伸阅读

• I. M. Gelfand and A. Shen, Algebra (1993)