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
A book-of-the-month club charges $13 for the first book purchased during the month and an additional $11 for each book after the first. Which equation describes this situation, where y is the total cost, in dollars, for books purchased during a month, x is the number of books purchased during the month, and \( x>0 \)?
原库参考解析
线性建模・建立方程挑战
A partially filled container containing 24 milliliters of water is placed under a leaky faucet that produces one 0.03-milliliter drop of water every 3 seconds. Until the container is full, which of the following can be used to represent the volume v, in milliliters, of water in the container t seconds after it is placed under the faucet, where t is a multiple of 3?
原库参考解析
线性建模・建立方程进阶
The gas mileage of a certain car is 22 miles per gallon when driven on the highway. If the car begins with 8 gallons of gas in the tank, which function predicts the amount of gas remaining in the tank, f (d), after the car is driven d miles on the highway?
原库参考解析
线性建模・建立方程进阶
Dan bought some apples and oranges. The apples cost $0.79 per pound, and the oranges cost $0.99 per pound. If Dan spent $24.93 in total and bought twice as many pounds of oranges as pounds of apples, how many pounds of oranges did Dan buy?
原库参考解析
线性建模・建立方程挑战
At a constant temperature, a scuba diver combines two different concentrations of nitrox mix (oxygen-nitrogen gas mixture) to fill a tank to a total pressure of 90 atmospheres (atm), of which 37% is contributed by oxygen. Nitrox mix A contributes x atm of pressure to the tank, of which 22% is contributed by oxygen. Nitrox mix B contributes y atm of pressure to the tank, of which 49% is contributed by oxygen. Which system of equations represents this situation?
原库参考解析
线性建模・建立方程进阶
The cost to rent a bus from Company X is $950 for the first 3 hours and an additional $50 per hour for each hour after the first 3 hours. If the total cost to rent the bus from Company X for t hours, where t > 3, is $1,100, which equation represents this situation?
原库参考解析
线性建模・建立方程挑战
The table shows the linear relationship between the number of cars, c, on a commuter train and the maximum number of passengers and crew, p, that the train can carry. Which equation represents the linear relationship between c and p?
原库参考解析
线性建模・建立方程进阶
A veterinarian recommends that each day a certain cat should eat 31 calories per pound of the cat's weight, plus an additional 14 calories. Which equation represents this situation, where c is the total number of calories the veterinarian recommends the cat should eat each day if the cat's weight is x pounds?
原库参考解析
线性建模・建立方程进阶
Javier deposits $45 in a savings account at the end of each week. At the beginning of the 1st week of a year there was $700 in that savings account. How much money, in dollars, will be in the account at the end of the 6th week of the year?
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 $64n for the first day and $32n 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・建立方程进阶
A computer repair specialist charges $160 for the first two hours of repair plus an hourly fee for each additional hour. The total cost for 4 hours of repair is $240. Which function f gives the total cost, in dollars, for x hours of repair; where x \( \ge \) 2?
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 $78n for the first day and $39n 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
