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 linear function \( f \) gives a company's profit, in dollars, for selling x items. The company's profit is $190 when it sells 4 items, and its profit is $670 when it sells 10 items. Which equation defines ?
原库参考解析
线性建模・建立方程挑战
The cost of renting a piece of construction equipment for up to 10 days is $230 for the first day and $115 for each additional day. Which of the following equations gives the cost y, in dollars, of renting the equipment for x days, where x is a positive integer and \( x\le10 \)?
原库参考解析
线性建模・建立方程挑战
The roofs on the buildings of a public garden are designed to allow water to drain into the garden. The table shows the relationship between the area x, in square feet, of a roof and the amount of water \( f\left(x\right) \), in gallons, drained from the roof into the garden over a certain period of time. Which equation could define \( f \)?
原库参考解析
线性建模・建立方程挑战
In 2010, a skating team had a total of 37 skaters, each classified as either advanced or intermediate. From 2010 to 2020, the number of advanced skaters on the team increased by approximately 23%, and the number of intermediate skaters on the team increased by approximately 33%. The total number of skaters on the team increased by approximately 27%. Which equation best represents this situation, where a represents the number of advanced skaters on the team in 2010 and b represents the number of intermediate skaters on the team in 2010?
原库参考解析
线性建模・建立方程挑战
The value of y is 8 more than twice the value of x. Which equation represents y in terms of x ?
原库参考解析
线性建模・建立方程挑战
A park rents beach umbrellas to visitors. The park earns revenue of $15 for each beach umbrella they rent for the day. On Wednesday, the park earned $435 in profit from renting beach umbrellas after paying daily expenses of $150. How many beach umbrellas did the park rent on Wednesday? (profit = total revenue - total expenses)
原库参考解析
线性建模・建立方程进阶
A machine makes 8-inch, 9-inch, and 3-inch parts. During a certain day, the number of 8-inch parts that the machine makes is 4 times the number n of 9-inch parts, and the number of 3-inch parts is 30. 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 linear function \( f \) gives a company's profit, in dollars, for selling x items. The company's profit is $170 when it sells 4 items, and its profit is $590 when it sells 10 items. Which equation defines ?
原库参考解析
线性建模・建立方程进阶
Casey is raising h horses and g goats. Each horse requires 2 acres of land, and each goat requires 1 acre of land. The h horses and g goats require a total of 21 acres of land. Which equation represents this situation?
原库参考解析
线性建模・建立方程进阶
A chemist mixed x liters of a 2% saline solution with y liters of a 5% saline solution to produce a 3% saline solution. Which equation best represents this situation? (Assume the volumes of the solutions are additive.)
原库参考解析
线性建模・建立方程挑战
The gas mileage of a certain car is 35 miles per gallon when driven on the highway. If the car begins with 10 gallons of gas in the tank, which function predicts the amount of gas remaining in the tank, \( f\left(d\right) \), after the car is driven d miles on the highway?
原库参考解析
线性建模・建立方程进阶
Raheem bought 9 shirts that were each the same price. He used a coupon for $54 off the entire purchase. The cost for the entire purchase after using the coupon was $108. What was the original price, in dollars, for 1 shirt?
原库参考解析
