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
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 3.8 micrograms of vitamin B12 are consumed from eating x ounces of cheese and y ounces of tuna, which equation best represents this situation?
参考答案
线性建模・建立方程挑战
As a car approaches an intersection, a linear function \( f \) models its speed, in miles per hour (mph), x seconds after the brake is initially applied. According to the function, the car's estimated speed is 23 mph when the brake is initially applied, and the car's estimated speed decreases by 4.1 mph every second after the brake is initially applied. Which equation defines \( f \)?
参考答案
线性建模・建立方程挑战
A set designer is hammering nails along the top edge of a wall with a length of \( 6x \) feet, where x is an integer, that will be used to hang garland. The designer hammers the first nail at the left edge of the wall. The designer then hammers an additional nail every \( \frac{6}{11} \) feet along the entire length of the wall, starting from the first nail and with the last nail being hammered at the right edge of the wall. Which equation best represents this situation, where y is the number of nails the designer hammers along the entire length of the wall?
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 45% discount to the towel and a 25% discount to the chair are applied, the combined sale price for the two items is $28. 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・建立方程进阶
Two candidates ran for the office of mayor of a town. A total of 4,500 votes were cast, and one candidate received twice as many votes as the other. How many votes did the winning candidate receive?
参考答案
线性建模・建立方程挑战
A moving truck rental company charges its customers $175 for the first hour of renting a truck and $65 for each additional hour, plus the cost of gas used. Which equation represents this situation, where y is the total cost, in dollars, of renting a truck from this company for x hours, w is the cost, in dollars, of gas used, and \( x\ge1 \)?
参考答案
线性建模・建立方程进阶
A moving truck rental company charges its customers $135 for the first hour of renting a truck and $85 for each additional hour, plus the cost of gas used. Which equation represents this situation, where y is the total cost, in dollars, of renting a truck from this company for x hours, w is the cost, in dollars, of gas used, and \( x\ge1 \)?
参考答案
线性建模・建立方程基础
During a 6-day period, a weather station recorded a total of 3.3 inches of rain and snow. During this period, the weather station recorded r inches of rain and s inches of snow. Which equation represents this situation?
参考答案
线性建模・建立方程挑战
A landscaping company mixes oil with gasoline to power its tools. A chainsaw requires 8 ounces of oil per gallon of gasoline. A lawnmower requires 4 ounces of oil per gallon of gasoline. The company uses a total of 24 ounces of oil in its mixes for the chainsaw and lawnmower. Which of the following equations represents this situation, where x represents the amount of gasoline, in gallons, in the mix for the chainsaw and y represents the amount of gasoline, in gallons, in the mix for the lawnmower?
参考答案
线性建模・建立方程挑战
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 $600 in that savings account. How much money, in dollars, will be in the account at the end of the 8th week of the year?
参考答案
线性建模・建立方程进阶
At a concession stand, lollipops cost $0.25 each and soft drinks cost $1.00 each. If Noah purchased 2 lollipops and some soft drinks for a total of $4.50, how many soft drinks did Noah purchase?
参考答案
线性建模・建立方程基础
A veterinarian recommends that each day a certain cat should eat 25 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?
参考答案
