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 computer repair specialist charges $160 for the first two hours of repair plus an hourly fee for each additional hour. The total cost for 6 hours of repair is $360. Which function \( f \) gives the total cost, in dollars, for x hours of repair, where \( x\ge2 \) ?
原库参考解析
线性建模・建立方程进阶
A club plans to sell tote bags. The club members estimate they will sell 80 tote bags when the bags are priced at $9 each. For every price increase of $1, they estimate they will sell 8 fewer bags. What is the estimated revenue, in dollars, when the bags are priced at $12 each? (revenue = price × number sold)
原库参考解析
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?
原库参考解析
线性建模・建立方程进阶
A gym charges its members a onetime $34 enrollment fee and a membership fee of $16 per month. If there are no charges other than the enrollment fee and the membership fee, after how many months will a member have been charged a total of $178 at the gym?
原库参考解析
线性建模・建立方程挑战
A carpenter charges a flat rate of $216 for the first 3 hours of work and $60 for each additional hour of work. Which equation gives the total amount y, in dollars, that the carpenter charges for x hours of work, where \( x>3 \) ?
原库参考解析
线性建模・建立方程挑战
The total cost \( f(x) \) , in dollars, to lease a car for 24 months from a particular car dealership is given by \( f(x)=24x+1,000 \) , where x is the monthly payment, in dollars. What is the total cost to lease a car when the monthly payment is $300?
原库参考解析
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 14 people per minute. At the end of a certain day, 238 people are in line for the ride when the line closes. If these 238 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?
原库参考解析
线性建模・建立方程进阶
A boat traveled a certain distance upriver and then traveled downriver 10 miles farther than it traveled upriver. The boat took \( x \) hours to travel upriver at a speed of 12 miles per hour. The boat then took \( y \) hours to travel downriver at a speed of 18 miles per hour faster than the speed at which the boat traveled upriver. Which of the following equations best represents this situation?
原库参考解析
线性建模・建立方程挑战
A business owner purchased an energy-efficient air conditioning system for a cost of $4,200, including installation. The business owner saves $940 per year to operate the new system compared to the operating cost of the old system. Including the cost of the new system, which function \( f \) gives the total amount, in dollars, the business owner has saved \( x \) years after purchasing the new system?
原库参考解析
线性建模・建立方程挑战
A machine makes 8-inch, 5-inch, and 7- inch parts. During a certain day, the number of 8-inch parts that the machine makes is 3 times the number n of 5-inch parts, and the number of 7-inch parts is 12. During this day, the machine makes 100 parts total. Which equation represents this situation?
原库参考解析
线性建模・建立方程挑战
A beekeeper's initial observation of the population of a certain bee colony was 1,100 bees. The beekeeper set a goal to increase the population to 2,600 bees. The beekeeper uses a model that predicts the population of this bee colony begins at 1,100 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?
原库参考解析
线性建模・建立方程进阶
Carrie has c dollars and Marjani has m dollars. Carrie has 4 times as many dollars as Marjani, and together they have a total of $40.00. Which system of equations represents this situation?
原库参考解析
