这部分需要会什么？
- 确定变量及单位
- 将固定量和单位变化量写成线性关系
- 从总量、差量和单价关系建立方程或方程组
核心概念与公式
线性模型 \(y=mx+b\) 中，\(b\) 是 \(x=0\) 时的初始量，\(m\) 是每单位 \(x\) 带来的固定变化量。若 \(x\) 的单位为小时、\(y\) 为千米，\(m\) 的单位就是千米/小时。
建立模型的关键是找到关系：总价等于单价乘数量；路程等于速度乘时间；两类物品的总数等于两类数量之和。只有相同时间内增减量固定时，才适合直接用线性模型。
做题时怎样使用
- 用一句话定义每个变量，写明单位
- 把总量、固定费用、每单位费用等分别列出
- 建立等式并检查一个简单输入，例如数量为零时是否符合题意
容易混淆的地方
初始量未必是零。每次增长固定百分比通常是指数变化，不能直接写成每次增加固定数量。
用一个例题串起方法
知识点教学例题租车有 30 元固定费用，另按每千米 2.4 元收费。写出行驶 \(d\) 千米的费用 \(C\)，并求 18 千米的费用。
展开解题思路 ＋
固定费用是截距，单位里程费用是斜率，所以 \(C=30+2.4d\)。当 \(d=18\) 时，\(C=73.2\) 元。
本知识点题库
先独立作答，再查看答案与解析。
线性建模・建立方程挑战
The cost of renting a power washer is $44 for the first day and $22 for each additional day. Which of the following functions gives the cost C(d) , in dollars, of renting the power washer for d days, where d is a positive integer?
原库参考解析
线性建模・建立方程挑战
The cost of renting a piece of construction equipment for up to 10 days is $370 for the first day and $185 for each additional day. Which of the following equations gives the cost y. in dollars, of renting the equipment for x days, where x is a positive integer and x ≤10?
原库参考解析
线性建模・建立方程挑战
Lao walks at a speed of 2 miles per hour and runs at a speed of 5 miles per hour. He walks for w hours and runs for r hours for a combined total of 11 miles. Which equation represents this situation?
原库参考解析
线性建模・建立方程挑战
The cost of renting a piece of construction equipment for up to 5 days is $390 for the first day and $195 for each additional day, Which of the following equations gives the cost y, in dollars, of renting the equipment for x days, where x is a positive integer and x \( \le \) 5?
原库参考解析
线性建模・建立方程挑战
The cost of renting a piece of construction equipment for up to 5 days is $390 for the first day and $195 for each additional day. Which of the following equations gives the cost y, in dollars, of renting the equipment for a days, where x is a positive integer and x ≤5?
原库参考解析
线性建模・建立方程进阶
In 2010, a skating team had a total of 43 skaters, each classified as either advanced or intermediate. From 2010 to 2020, the number of advanced skaters on the team decreased by approximately 15%, and the number of intermediate skaters on the team decreased by approximately 22%. The total number of skaters on the team decreased by approximately 19%. Which equation best represents this situation, where a represents the number of advanced skaters on the team in 2010 and b represents the number of intermediate skaters on the team in 2010?
原库参考解析
线性建模・建立方程挑战
A business consultant charges $408 for the first hour and $204 for each additional hour of work. Which of the following functions gives the charge C(h), in dollars, for h hours of work, where h is a positive integer?
原库参考解析
线性建模・建立方程挑战
A business consultant charge $408 for the first hour and $204 for each additional hour of work. Which of the following functions gives the charge C(h), in dollars, for h hours of work, where h is a positive integer?
原库参考解析
线性建模・建立方程挑战
The cost of renting a piece of equipment is $46n for the first day and $23n for each additional day, where n is a positive integer Which of the following functions gives the cost C(x), in dollars, of renting this equipment for x days, where x is a positive integer ?
原库参考解析
线性建模・建立方程挑战
During a specific 2-hour period of the day, the mean photosynthetic rate, P(x), of a plant species increases at a constant rate in terms of the photochemical energy available, x. During this period, P(300) = 9.5, and the mean photosynthetic rate increases by 0.125 micromoles per meter squared per second (\( μ \) mol • \( m^{-2}s^{-1} \) ) when the photochemical energy available increases by 5\( μ \) mol • \( m^{-2}s^{-1} \). Which equation gives the mean photosynthetic rate, in \( μ \) mol • \( m^{-2}s^{-1} \), in terms of the photochemical energy available, in \( μ \) mol • \( m^{-2}s^{-1} \), during this period?
原库参考解析
线性建模・建立方程进阶
The height of a certain vehicle is 61 inches. The vehicle's length is 3 times its height. Which of the following systems of equations represents this situation, where x is the length of the vehicle, in inches, and y is the height of the vehicle, in inches?
原库参考解析
线性建模・建立方程挑战
For groups of 25 or more people, a museum charges $28 per person for the first 25 people and $17 for each additional person. Which function f gives the total charge, in dollars, for a tour group with n people, where \( n\ge25 \) ?
原库参考解析
