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 $16 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
Micah deposited a total of $6,600 into two retirement accounts last year by depositing x dollars once each month into one account and y dollars twice each month into the other account. Which equation represents this situation?
原库参考解析
线性建模・建立方程挑战
A book-of-the-month club charges $16 for the first book purchased during the month and an additional $12 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
Ellie deposited a total of $4,300 into two retirement accounts last year by depositing x dollars once each month into one account and y dollars twice each month into the other account. Which equation represents this situation?
原库参考解析
线性建模・建立方程挑战
A store sells bottles of water for $0.93 each and makes a profit of $0.35 on each bottle sold. During the first three days of a week, the store sold 128 bottles of water. The rest of the week, the store sold w bottles of water. The store made a total profit of $130.20 from selling bottles of water during the week. Which equation represents this situation?
原库参考解析
线性建模・建立方程进阶
There are a total of 1,200 milligrams of phosphorus in a sample of soil. Each kilogram of this sample contains 600 milligrams of phosphorus. Which equation represents this situation, where x is the mass, in kilograms, of the sample of soil?
原库参考解析
线性建模・建立方程挑战
The cost to rent a kayak consists of a fixed fee for the first hour and an hourly fee for each additional hour. The table shows the rental cost for 4 hours and for 6 hours.
Which function f gives the rental cost, in dollars, for x hours rental, where \( x\ge1 \)?
原库参考解析
线性建模・建立方程挑战
Each 3.0-ounce serving of cheddar cheese and each 1.2-ounce serving of tuna provides about 1 microgram of vitamin B12. If a total of 1.7 micrograms of vitamin B12 are consumed from eating x ounces of cheese and y ounces of tuna, which equation best represents this situation?
原库参考解析
线性建模・建立方程挑战
A partially filled container containing 21 milliliters of water is placed under a leaky faucet that produces one 0.06-milliliter drop of water every 6 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 6?
原库参考解析
线性建模・建立方程挑战
Hannah and Wyatt are saving money to purchase a car. Hannah saves \( \frac{1}{5} \) of her salary each month, and Wyatt saves \( \frac{2}{7} \) of his salary each month. Together, they save a total of $3,270 from their monthly salaries each month. If h and w represent Hannah's and Wyatt's monthly salaries, in dollars, respectively, which equation shows the relationship between h and w ?
原库参考解析
线性建模・建立方程进阶
The length, y, of a beluga whale was 158 centimeters (cm) when it was born and increased an average of 4.9 cm per month for the first 12 months after it was born. Which equation best represents this situation, where x is the number of months after the whale was born?
原库参考解析
线性建模・建立方程挑战
A wireless phone company pays customers for trading in old smartphones. The payment p, in dollars, is the original retail price of a smartphone minus $135 for each year since the smartphone was released. Which equation represents this situation for a smartphone with an original retail price of $750 when it was released t years ago, where t is a whole number and \( 1\le t\le4 \)?
原库参考解析
