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・建立方程进阶
The cost to rent a bus from Company X is $950 for the first 3 hours and an additional $50 per hour for each hour after the first 3 hours. If the total cost to rent the bus from Company X for t hours, where \( t>3 \), is $1,150, which equation represents this situation?
原库参考解析
线性建模・建立方程进阶
\( 8x+11y=170 \)
The equation gives the possible combinations of the number of 2009 premium grade Log Cabin Pennies, x, and the number of 1996 select grade Lincoln Pennies, y, in a collection that is worth a total of $170. If there are 6 1996 select grade Lincoln Pennies in the collection, how many 2009 premium grade Log Cabin Pennies are in the collection?
原库参考解析
线性建模・建立方程基础
In a chess tournament, each participant earns 1 point for each game the participant plays that ends in a draw and 3 points for each game the participant wins. A certain participant in this tournament has earned 41 points. Which equation represents this situation, where d represents the number of games this participant has played that ended in a draw and w represents the number of games this participant has won?
原库参考解析
线性建模・建立方程挑战
Josie purchased a pass that cost $19.50 to tour a nature preserve. She also purchased 10 identical stickers to give to her nieces and nephews. The pass and the stickers cost $62.00 total. What was the cost of one sticker?
原库参考解析
线性建模・建立方程进阶
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・建立方程基础
The total cost f(x), in dollars, to lease a car for 36 months from a particular car dealership is given by f(x) = 36x ＋ 1,000, where x is the monthly payment, in dollars. What is the total cost to lease a car when the monthly payment is $400?
原库参考解析
线性建模・建立方程基础
John paid a total of $165 for a microscope by making a 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・建立方程基础
The number y is 84 less than the number x. Which equation represents the relationship between x and y?
原库参考解析
线性建模・建立方程挑战
For groups of 25 or more people, a museum charges $21 per person for the first 25 people and $14 for each additional person. Which function f gives the total charge, in dollars, for a tour group with n people, where n ≥ 25?
原库参考解析
线性建模・建立方程基础
3 more than 8 times a number x is equal to 83.Which equation represents this situation?
原库参考解析
线性建模・建立方程基础
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・建立方程基础
Nasir bought 9 storage bins that were each the same price.He used a coupon for $63 off the entire purchase. The cost for the entire purchase after using the coupon was $27.What was the original price, in dollars, for 1 storage bin?
原库参考解析
