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・建立方程基础
A city has an area of x square miles and a population density of 1,035 people per square mile. The suburb adjacent to the city has an area of y square miles and a population density of 535 people per square mile. The combined population of the city and the suburb adjacent to the city is 13,165 people. Which equation represents this situation?
原库参考解析
线性建模・建立方程进阶
During hibernation, American black bears do not eat or replenish calories. A certain black bear weighed 293 pounds when entering hibernation and lost weight at a mean rate of 0.9 pounds per day. At this rate, how many days after entering hibernation would the black bear weigh 230 pounds?
原库参考解析
线性建模・建立方程挑战
A biologist mixed a solution that is 0.3% sodium chloride by mass with a solution that is 0.15% sodium chloride by mass to obtain a new solution, which has a mass of 80 grams and contains 0.21 grams of sodium chloride. How many grams of 0.3% sodium chloride solution did the biologist use?
原库参考解析
线性建模・建立方程挑战
At a fundraiser, a family pays a fixed entrance fee. They then purchase food tickets, which each have the same cost. The table shows the relationship between the number of food tickets, x, that are purchased and the total amount paid y, in dollars, for both the entrance fee and the food tickets. Which equation represents the relationship between x and y?
原库参考解析
线性建模・建立方程进阶
A beekeeper’s initial observation of the population of a certain bee colony was 1,600 bees. The beekeeper set a goal to increase the population to 2,975 bees. The beekeeper uses a model that predicts the population of this bee colony begins at 1,600 and increases by 110 bees per week in the first two weeks after the initial observation, and then increases by 165 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?
原库参考解析
线性建模・建立方程进阶
A dance studio charges an introductory fee for the first 3 lessons and then charges a fixed fee for each additional lesson. One student took 4 lessons and was charged $60. Another student took 16 lessons and was charged $300. Which function \( f \) gives the total charge, in dollars, for any student who took x lessons, where \( x>2 \)?
原库参考解析
线性建模・建立方程进阶
Hector receives a gift card worth $70.00 to pay for movies online. He uses his gift card to buy 2 movies for $6.50 each. He will spend the rest of his gift card on renting movies for $1.50 each. How many movies can Hector rent?
原库参考解析
线性建模・建立方程挑战
When x liters of a 40% acid solution is mixed with y liters of a 60% acid solution, 12 liters of a 50% acid solution is produced. Which system of equations represents this situation? (Assume that the volume of the mixture is the sum of the volumes of the two solutions before they were mixed.)
原库参考解析
线性建模・建立方程进阶
A cell phone plan costs 82 dollars each month. Which equation gives the total cost c, in dollars, of the cell phone plan for m months?
原库参考解析
线性建模・建立方程挑战
A shipment consists of 5-pound boxes and 10-pound boxes with a total weight of 360 pounds. There are 23 10-pound boxes in the shipment. How many 5-pound boxes are in the shipment?
原库参考解析
线性建模・建立方程进阶
Based on a model, on day 1 of an 8-day experiment, a lima bean plant had an estimated mass of 253 grams, and each day after day 1 of the experiment, the estimated mass of the plant decreased by 4 grams. Which equation represents this model, where m is the estimated mass, in grams, of the plant on day x of the experiment and \( 1\le x\le8 \)?
原库参考解析
线性建模・建立方程基础
A total of 255 people registered for a city's annual race as either a runner or a walker. 210 people registered as a runner. How many people registered as a walker?
原库参考解析
