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・建立方程进阶
The price to rent an event space is 1,050 dollars for the first 3 hours and 250 dollars for each additional hour. What is the total price, in dollars, to rent the event space for 10 hours?
参考答案
线性建模・建立方程挑战
A machine makes 4-inch, 7-inch, and 9-inch parts. During a certain day, the number of 4-inch parts that the machine makes is 5 times the number n of 7-inch parts, and the number of 9-inch parts is 4. During this day, the machine makes 100 parts total. Which equation represents this situation?
参考答案
线性建模・建立方程进阶
A beekeeper's initial observation of the population of a certain bee colony was 1,800 bees. The beekeeper set a goal to increase the population to 3,300 bees. The beekeeper uses a model that predicts the population of this bee colony begins at 1,800 and increases by 120 bees per week in the first two weeks after the initial observation, and then increases by 180 bees per week until the beekeeper's goal is reached. According to this model, at the end of week w after the initial observation, where \( w>2 \), which of the following functions gives the predicted number of bees still needed to reach the beekeeper's goal?
参考答案
线性建模・建立方程挑战
The combined original price for a towel and chair is $40. After a 35% discount to the towel and a 15% discount to the chair are applied, the combined sale price for the two items is $31. Which system of equations gives the original price t, in dollars, of the towel and the original price c, in dollars, of the chair?
参考答案
线性建模・建立方程挑战
A biologist mixed a solution that is 0.1% sodium chloride by mass with a solution that is 0.05% sodium chloride by mass to obtain a new solution, which has a mass of 90 grams and contains 0.07 grams of sodium chloride. How many grams of the 0.1% sodium chloride solution did the biologist use?
参考答案
线性建模・建立方程挑战
At an amusement park, visitors wait in line for a certain ride. People in this line get on the ride at a rate of 12 people per minute. At the end of a certain day, 201 people are in line for the ride when the line closes. If these 201 people will get on the ride and no additional people are allowed to get in line, which function p models the number of people who remain in line t minutes after it closes?
参考答案
线性建模・建立方程挑战
A beekeeper’s initial observation of the population of a certain bee colony was 1,800 bees. The beekeeper set a goal to increase the population to 3,300 bees. The beekeeper uses a model that predicts the population of this bee colony begins at 1,800 and increases by 120 bees per week in the first two weeks after the initial observation, and then increases by 180 bees per week until the beekeeper’s goal is reached. According to this model, at the end of week w after the initial observation, where \( w>2 \), which of the following functions gives the predicted number of bees still needed to reach the beekeeper’s goal?
参考答案
线性建模・建立方程挑战
A manager is responsible for ordering supplies for a coffee shop. The shop's inventory starts with 6,500 disposable cups, and the manager estimates that 90 of these cups are used each day. Based on this estimate, in how many days will the supply of disposable cups reach 1,100?
参考答案
线性建模・建立方程挑战
Each side of a 30-sided polygon has one of three lengths. The number of sides with length 4 centimeters (cm) is 5 times the number of sides n with length 8 cm. There are 6 sides with length 3 cm. Which equation must be true for the value of n?
参考答案
线性建模・建立方程进阶
Maya bought stickers and granola bars for $145. Each package of stickers cost $3, and each granola bar cost $1. If Maya bought 25 packages of stickers, how many granola bars did she buy?
参考答案
线性建模・建立方程挑战
During the month of May, an airport parking garage is offering a special price plan. According to this plan, the cost of parking in the garage for up to 5 days is $18 for the first day and $9 for each additional day. Which of the following equations gives the cost C, in dollars, of parking in the garage for t days, where t is a positive integer and \( t\le5 \)?
参考答案
线性建模・建立方程进阶
Quinidra drove her car on a straight road to a certain location. The graph models Quinidra’s distance, in miles, from her home as a function of time, in hours. Which of the following could the model describe?
参考答案
