Linear relationships and simultaneous equations

前置知识

14 项必需的前置知识（一直追溯到底）

你将学会

• Compute the slope of a line from two points and interpret it as a rate of change.
• Write the equation of a line from its slope and a point, and recognise linear functions in tables, graphs and formulas.
• Solve two simultaneous linear equations by substitution, elimination and graphing, and interpret the cases with no solution or infinitely many.

定义

• The slope of the line through (x₁, y₁) and (x₂, y₂), with x₁ ≠ x₂, is (y₂ − y₁)/(x₂ − x₁).
• A linear function has the form f(x) = mx + b.
• A solution of a system of equations satisfies every equation in the system at once.

定理

• The slope between any two points on a non-vertical line is the same.
• Two distinct lines in the plane meet in exactly one point unless they are parallel, that is, have equal slopes or are both vertical.
• A system of two linear equations in two unknowns has exactly one solution, no solution, or infinitely many solutions.

常见误解

• Computing slope as 'run over rise'.
• Believing every straight-line relationship is proportional.
• Giving a vertical line slope zero; its slope is undefined.

通向

可选发现

• Function machines in sequence Composing two linear functions gives another: applying x ↦ m₂x + b₂ and then x ↦ m₁x + b₁ gives x ↦ m₁m₂x + (m₁b₂ + b₁).

学习路径

• Conventional K–12 progression

延伸阅读

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