知识点地图 / 代数

Building Linear Models

线性建模・建立方程

识别固定量和单位变化量,将文字条件转化成一次方程或方程组。

01 / UNDERSTAND

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

  1. 用一句话定义每个变量,写明单位
  2. 把总量、固定费用、每单位费用等分别列出
  3. 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。

02 / CONNECT

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。

03 / PRACTISE

本知识点题库

先独立作答,再查看答案与解析。

202 道题目
找到 202 道题目重置筛选
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 \( \ge \) 25 ?
原库参考解析
Linear functions · 原库模块 1 · Q15作答 ↗
线性建模・建立方程挑战

A window repair specialist charges $220 for the first two hours of repair plus an hourly fee for each additional hour. The total cost for 5 hours of repair is $400. Which function f gives the total cost, in dollars, for x hours of repair, where x \( \ge \) 2?
原库参考解析
Linear functions · 原库模块 1 · Q1作答 ↗
线性建模・建立方程挑战

The cost of renting a backhoe for up to 10 days is $270 for the first day and $135 for each additional day. Which of the following equations gives the cost y, in dollars, of renting the backhoe for x days, where x is a positive integer and x \( \le \) 10?
原库参考解析
Linear functions · 原库模块 1 · Q3作答 ↗
线性建模・建立方程进阶

The length, y, of a white whale was 162 centimeters (cm) when it was born and increased an average of 4.8 cm per month for the first 12 months after it was born. Which equation best represents this situation, where x is the number of months after the whale was born and y is the length, in cm, of the whale?
原库参考解析
PSAT 8/9 P1 · 原库模块 1 · Q2作答 ↗
线性建模・建立方程进阶

Connor has c dollars and Maria has m dollars. Connor has 4 times as many dollars as Maria, and together they have a total of $25.00. Which system of equations represents this situation?
原库参考解析
PSAT 8/9 P1 · 原库模块 1 · Q6作答 ↗
线性建模・建立方程基础

A shipment consists of 5-pound boxes and 10-pound boxes with a total weight of 220 pounds. There are 13 10-pound boxes in the shipment. How many 5-pound boxes are in the shipment?
原库参考解析
PSAT 8/9 P1 · 原库模块 2 · Q15作答 ↗
线性建模・建立方程基础

An employee at a restaurant prepares sandwiches and salads. It takes the employee 1.5 minutes to prepare a sandwich and 1.9 minutes to prepare a salad. The employee spends a total of 46.1 minutes preparing x sandwiches and y salads. Which equation represents this situation?
原库参考解析
PSAT 8/9 P1 · 原库模块 2 · Q8作答 ↗
线性建模・建立方程挑战

A gym charges its members a onetime $36 enrollment fee and a membership fee of $19 per month. If there are no charges other than the enrollment fee and the membership fee, after how many months will a member have been charged a total of $188 at the gym?
原库参考解析
PSAT 8/9 P1 · 原库模块 3 · Q1作答 ↗
线性建模・建立方程进阶

Gabriella deposits $35 in a savings account at the end of each week. At the beginning of the 1st week of a year there was $600 in that savings account. How much money, in dollars, will be in the account at the end of the 4th week of that year?
原库参考解析
PSAT 8/9 P2 · 原库模块 1 · Q12作答 ↗
线性建模・建立方程进阶

To repair a refrigerator, a technician charges $60 per hour for labor plus $120 for parts. Which function \( f \) represents the total amount, in dollars, the technician will charge for this job if it takes x hours?
原库参考解析
PSAT 8/9 P2 · 原库模块 1 · Q2作答 ↗
线性建模・建立方程基础

Naomi bought both rabbit snails and nerite snails for a total of $52. Each rabbit snail costs $8 and each nerite snail costs $6. If Naomi bought 2 nerite snails, how many rabbit snails did she buy?
原库参考解析
PSAT 8/9 P2 · 原库模块 2 · Q13作答 ↗
线性建模・建立方程基础

A food truck buys forks for $0.04 each and plates for $0.48 each. The total cost of x forks and y plates is $661.76. Which equation represents this situation?
原库参考解析
PSAT 8/9 P2 · 原库模块 2 · Q16作答 ↗