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
For groups of 6 or more people, a tour guide charges $14 for the first person in the group and $12 for each additional person. Which function g gives the total charge, in dollars, for a tour with a group of n people, where \( n\ge6 \)?
原库参考解析
线性建模・建立方程进阶
A large square has an area of 1,296 square centimeters (cm²). Small squares of area 36 cm² are cut out of the large square one by one. Which function f gives the area of the resulting figure, in cm², after x small squares are cut out?
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 4,880 cubic centimeters (cm³) of sand and cement, where the ratio of the volume of sand to the volume of cement is 7 to 3. If s represents the volume, in cm³, of sand and c represents the volume, in cm³, 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 set designer is hammering nails along the top edge of a wall with a length of 3x feet, where x is an integer, that will be used to hang garland. The designer hammers the first nail at the left edge of the wall. The designer then hammers an additional nail every \( \frac{3}{10} \) feet along the entire length of the wall, starting from the first nail and with the last nail being hammered at the right edge of the wall. Which equation best represents this situation, where y is the number of nails the designer hammers along the entire length of the wall?
原库参考解析
线性建模・建立方程挑战
The cost of renting a power washer, is $76 for the first day and $38 for each additional day. Which of the following functions gives the cost C(d) , in doilars, of renting the power washer for d days, where d is a pasitive integer?
原库参考解析
线性建模・建立方程挑战
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 hest represents this situation? (Assume the volumes of the solutions are additive.)
原库参考解析
线性建模・建立方程进阶
A landscaping company mixes oil with gasoline to power its tools. A chainsaw requires 8 ounces of oil per gallon of gasoline. A lawnmower requires 4 ounces of oil per gallon of gasoline. The company uses a total of 21 ounces of oil in its mixes for the chainsaw and lawnmower. Which of the following equations represents this situation, where x represents the amount of gasoline, in gallons, in the mix for the chainsaw and y represents the amount of gasoline, in gallons, in the mix for the lawnmower?
原库参考解析
线性建模・建立方程挑战
A polygonal chain is a connected series of line segments whose length is defined as the sum of the lengths of the line segments it is made of. A polygonal chain with a length of 33 centimeters (cm) is extended by adding line segments that each have a length of \( 8−r \) cm. Which function \( f \) represents the total length, in cm, of the extended polygonal chain after x line segments are added?
原库参考解析
线性建模・建立方程挑战
A chemist combines 30 grams of compound A, 45 grams of compound B, and x grams of compound C. The resulting mixture has a mass of 135 grams. What is the value of x?
原库参考解析
线性建模・建立方程挑战
The table gives the number of hours, \( h \), of labor and a plumber's total charge \( f\left(h\right) \), in dollars, for two different jobs.
There is a linear relationship between \( h \)and \( f\left(h\right) \). Which equation represents this relationship?
原库参考解析
线性建模・建立方程挑战
A carnival receives $10 for each adult admission ticket sold and $5 for each child admission ticket sold. On a particular day, the carnival received a total of $4,100 from selling adult and child admission tickets. If 220 adult admission tickets were sold on that day, how many child admission tickets were sold?
原库参考解析
线性建模・建立方程挑战
Hassan put up wire fencing along each edge of a rectangular garden with a length of x feet and a width of y feet. Hassan put up a total length of 44 feet of wire fencing. Which equation represents this situation?
原库参考解析
