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 carpenter charges a flat rate of $228 for the first 2 hours of work and $95 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 > 2?
原库参考解析
线性建模・建立方程挑战
The cost of renting a power washer is $56 for the first day and $28 for each additional day. Which of the following functions gives the cost C(d), in dollars, of renting the power washer for d days, where d is a positive integer?
原库参考解析
线性建模・建立方程挑战
Lao walks at a speed of 2 miles per hour and runs at a speed of 6 miles per hour. He walksfor w hours and runs for r hours for a combined total of 10 miles. Which equation represents this situation?
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 5 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 ≤5?
原库参考解析
线性建模・建立方程进阶
An aquarium uses small and large tanks to temporarily hold a total of 87 fish. Each small tank holds 1 fish, and each large tank holds 2 fish. Which equation represents this situation, where s is the number of small tanks and b is the number of large tanks the aquarium uses?
原库参考解析
线性建模・建立方程挑战
A scientist makes batches of bricks using different sand-to-cement ratios to test how the sand-to-cement ratio affects the strength of the brick. For one batch of bricks, the scientist uses 4,890 cubic centimeters (\( cm^3 \) ) of sand and cement, where the ratio of the volume of sand to the volume of cement is 4 to 1. If s represents the volume, in \( cm \)\( ^{3} \), of sand and c represents the volume, in \( cm \)\( ^{3} \), of cement that the scientist uses for this batch of bricks, which system of equations represents this situation?
原库参考解析
线性建模・建立方程挑战
A certain elephant calf weighed 226 pounds when it was born. The elephant calf gained an average of 3 pounds per day in its first year of life. Which equation models this situation, where y is the estimated weight of the elephant calf, in pounds, x days after it was born?
原库参考解析
线性建模・建立方程挑战
A cooking school is offering a promotion where the first class is free, the second class is half of the regular price, and the remaining classes are regularly priced. If the regular price of a class is $22.80, which function \( f \) gives the total cost, in dollars, of x classes taken using this promotion, where \( x\ge2 \) ?
原库参考解析
线性建模・建立方程挑战
A painting studio is offering a promotion where the first class is free, the second class is half off the regular price, and the remaining classes are regularly priced. If the regular price of a class is $26.40, which function \( f \) gives the total cost, in dollars, of x classes taken using this promotion, where \( x\ge2 \) ?
原库参考解析
线性建模・建立方程进阶
Tanya drove on the highway and on local roads to complete a trip of 290 miles. The drive took 5 hours. She drove an average speed of 65 miles per hour (mph) on the highway and an average speed of 30 mph on local roads. If x is the time, in hours, Tanya drove on the highway and y is the time, in hours, she drove on local roads. Which system of equations represents this situation?
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 6 lessons and was charged $150. Another student took 17 lessons and was charged $480. 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
A group of students purchase ice cream cones and ice cream sundaes. The cones cost $1.40 each, and the cost for 19 sundaes is $64.60. The total cost for the cones and sundaes is $85.60. Which equation represents this situation, where x is the number of cones purchased and y is the number of sundaes purchased?
原库参考解析
