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 bus traveled on the highway and on local roads to complete a trip of 160 miles. The trip took 4 hours. The bus traveled at an average speed of 55 miles per hour mph on the highway and an average speed of 25 mph on local roads. If x is the time, in hours, the bus traveled on the highway and y is the time, in hours, it traveled on local roads, which system of equations represents this situation?
原库参考解析
线性建模・建立方程基础
For a camping trip a group bought x one-liter bottles of water and y three-liter bottles of water, for a total of 240 liters of water. Which equation represents this situation?
原库参考解析
线性建模・建立方程基础
Henry receives a $60.00 gift card to pay for movies online. He uses his gift card to buy 3 movies for $7.50 each. If he spends the rest of his gift card balance on renting movies for $1.50 each, how many movies can Henry rent?
原库参考解析
线性建模・建立方程基础
A contract for a certain service requires a onetime activation cost of $35 and a monthly cost of $23. Which equation represents this situation, where c is the total cost, in dollars, of this service contract for t months?
原库参考解析
线性建模・建立方程进阶
A teacher is creating an assignment worth 70 points. The assignment will consist of questions worth 1 point and questions worth 3 points. Which equation represents this situation, where x represents the number of 1-point questions and y represents the number of 3-point questions?
原库参考解析
线性建模・建立方程挑战
For a training program, Juan rides his bike at an average rate of 5.7 minutes per mile. Which function m models the number of minutes it will take Juan to ride x miles at this rate?
原库参考解析
线性建模・建立方程挑战
Jay walks at a speed of 3 miles per hour and runs at a speed of 5 miles per hour. He walks for w hours and runs for r hours for a combined total of 14 miles. Which equation represents this situation?
原库参考解析
线性建模・建立方程进阶
A candle is made of 17 ounces of wax. When the candle is burning, the amount of wax in the candle decreases by 1 ounce every 4 hours. If 6 ounces of wax remain in this candle, for how many hours has it been burning?
原库参考解析
线性建模・建立方程进阶
John paid a total of $165 for a microscope by making down payment of $37 plus p monthly payments of $16 each. Which of the following equations represents this situation?
原库参考解析
线性建模・建立方程进阶
3 more than 8 times a number x is equal to 83. Which equation represents this situation?
原库参考解析
