六至八年级Algebra计划中
Linear relationships and simultaneous equations
Lines in the coordinate plane: slope as a constant rate of change, the form y = mx + b, linear functions, and solving pairs of simultaneous linear equations graphically and algebraically.
math.algebra.linear-relationships
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前置知识
Linear equations and inequalitiesFunctionsThe coordinate planeRatios, rates and proportional reasoning
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
延伸阅读
- I. M. Gelfand and A. Shen, Algebra (1993)