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・建立方程基础
Kayra spent $74 at a book sale and bought a total of 22 books. She bought h hardcover books for $4 each and p paperback books for $3 each. 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 14 lessons and was charged $390. 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 manager is responsible for ordering supplies for a coffee shop. The shop's inventory starts with 6,600 disposable cups, and the manager estimates that 110 of these cups are used each day. Based on this estimate, in how many days will the inventory of disposable cups reach 1,100?
原库参考解析
线性建模・建立方程进阶
A chemist mixed x liters of a 4% saline solution with y liters of a 7% saline solution to produce a 6% 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 table shows the linear relationship between the number of cars, c, on a commuter train and the maximum number of passengers and crew, p, that the train can carry. Which equation represents the linear relationship between c and p?
原库参考解析
线性建模・建立方程进阶
A book publishing company pays the author of a certain book $3.75 per book for the first 400 books sold. After the first 400 books are sold, the payment increases to $4.50 per book sold. Which function gives the author's total payment \( P(b) \) , in dollars, in terms of the number of books sold, b, where \( b>400 \)?
原库参考解析
线性建模・建立方程进阶
\( f\left(x\right)=35x+500 \)
The function \( f \) gives the monthly fee \( f(x) \) in dollars, a storage company charges to keep x boxes in its warehouse. What is the monthly fee, in dollars, the company charges to keep 40 boxes in its warehouse?
原库参考解析
线性建模・建立方程挑战
Each 3.0-ounce serving of cheddar cheese and each 1.2-ounce serving of tuna provides about 1 microgram of vitamin B12. If a total of 3.2 micrograms of vitamin B12 are consumed from eating x ounces of cheese and y ounces of tuna, which equation best represents this situation?
原库参考解析
线性建模・建立方程进阶
Natalie bought both neon tetra fish and swordtail fish for a total of $42. Each neon tetra fish costs $5 and each swordtail fish costs $4. If Natalie bought 3 swordtail fish, how many neon tetra fish did she buy?
原库参考解析
线性建模・建立方程进阶
A botanist studied the growth rate of a birch tree over two years. In the first year, the height of the tree increased c inches. In the second year, the height of the tree increased d inches. The height of the tree increased a total of 48 inches over these two years. Which equation represents this situation?
原库参考解析
线性建模・建立方程挑战
An artist made 14 pieces of pottery using 48 pounds of clay. The pottery included only bowls and vases. Each bowl used 3 pounds of clay and each vase used 5 pounds of clay. How many vases did the artist make?
原库参考解析
线性建模・建立方程进阶
At a fundraiser, a family pays a fixed entrance fee. They then purchase food tickets, which each have the same cost. The table shows the relationship between the number of food tickets, x , that are purchased and the total amount paid y in dollars, for both the entrance fee and the food tickets. Which equation represents the relationship between x and y?
参考答案
