3.4 线性平价和分配财产

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  线性等义和分配财产

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  Selling Shirts
  ::销售衬衫

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  Patrick Duran is the CEO of Pronto Custom Shirts, a business that makes custom-designed shirts for events. This past year Pronto Custom Shirts had a net income of $334,000. Net income is how much money a business takes in after all expenses. Patrick sets a net income goal of $400,000 for next year. He sells his shirts for $15, and each shirt costs $6.50 to make on average considering the variable costs of buying the un-printed shirt, the labor, etc. The fixed expenses, including the screen printing machine, rent, air conditioning, among others, total $127,000. He is using the following equation to determine net income:
  ::Patrick Duran是Pronto Custopic Shilts的首席执行官,该企业为事件制作定制衬衫。去年,Pronto Bustom Shilts的净收入为334 000美元。净收入是企业在扣除所有开支后所得收入的多少。Patrick为明年设定了400 000美元的净收入目标。他以15美元的价格出售衬衫,每件衬衫平均花费6.50美元,以计算购买未印衬衫、劳动力等的可变成本。固定费用,包括屏幕印刷机、租金、空调等,共计127 000美元。他使用以下方程式确定净收入:

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  $\begin{array}{r}\text{Net Income = Sales Price}\cdot \text{Quantity Sold - Cost Per Shirt}\cdot \text{Quantity Sold - Fixed Costs}\end{array}$
  ::净收入 = 销售价格 = 销售额 = 销售额 = 销售额 = 销售额 = 销售额 = 销售额 = 销售额 = 销售额 = 销售额 = 销售额

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  Patrick wants to know how many shirts he will have to sell to reach this goal.
  ::Patrick想知道他要卖多少件衬衫才能达到这个目标

   

  

  INTERACTIVE

  Selling Shirts

  

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  • Use the buttons to increase and decrease the values. 
    ::使用按钮来增加和减少值。
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  • See if you can reach Nate's goal of a net income of $400,000.
    ::看看你能不能达到内特的目标 净收入40万美元

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  Patrick does not want to change his cost per shirt and sales prices. He also cannot change his fixed costs. Using the interactive, how many shirts will Patrick need to sell to reach his goal of $400,000 net income?
  ::Patrick不想改变每件衬衫的成本和销售价格。 他也不能改变固定成本。 利用互动,Patrick要卖多少衬衫才能达到40万美元净收入的目标?

  a

  65,000

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  a number between 60,000 and 65,000
  ::a 数字在60 000至65 000之间。

  c

  60,000

  d

  50,000

   

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  Discussion Question
  ::讨论问题

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  What are some ways that Patrick can reach his sales goal?
  ::Patrick有什么方法能达到他的销售目标?

   

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  Combining Like Terms
  ::将类似术语合并

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  Patrick wants to know how many shirts he will have to sell to reach this goal without changing the sales price or costs. The challenge faced by the owner of Pronto Custom Shirts can be solved using a multi-step equation. A   multi-step equation   is an equation that requires more than two steps to solve for the variable. These equations often involve combining like terms and " data-term="Parentheses" role="term" tabindex="0"> parentheses . In this example, you will be dealing with like terms. To solve an example with like terms, combine the like terms and solve the resulting equation.
  ::Patrick想知道他要达到这个目标就必须卖多少件衬衫才能不改变销售价格或成本。 Pronto Bustom Shitts所有者面临的挑战可以通过一个多步方程来解决。多步方程是需要两个以上步骤才能解决变量的方程。这些方程通常包括类似条件和括号的组合。在此示例中,您将处理类似条件。要用类似条件解决一个示例,将类似条件结合起来,将类似条件结合起来,并解决由此产生的方程。

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  B egin by  clearly stating  the information you have.
  ::首先要明确说明你掌握的信息

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  $\begin{array}{r}\bullet \end{array}$  Desired Net Income = $400,000
  ::* 希望净收入=400 000美元

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  $\begin{array}{r}\bullet \end{array}$  Sales Price = $15
  ::* 销售价格=15美元

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  $\begin{array}{r}\bullet \end{array}$  Cost Per Shirt = $6.50
  ::* 每件衣服的费用=6.50美元

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  $\begin{array}{r}\bullet \end{array}$  Fixed Costs = $127,000
  ::* 固定费用=127 000美元

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  $\begin{array}{r}\bullet \end{array}$  Quantity Sold =  $\begin{array}{r}q\end{array}$
  ::* 销售数量=q

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  When you substitute these into the equation from the introduction,  you get the following:
  ::当您用引言中的方程式替换这些方程式时,您可以得到以下信息:

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  $$\begin{array}{r}400000=15q-6.5q-127000\end{array}$$

  
  ::400000=15q-6.5q-127000

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  Combining the like terms  $\begin{array}{r}15q\end{array}$  and  $\begin{array}{r}-6.5q\end{array}$  results in  $\begin{array}{r}8.5q.\end{array}$  
  ::将类似条件15q和-6.5q合并得出8.5q。

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  $$\begin{array}{r}400000=8.5q-127000\end{array}$$

  
  
  ::400000=8.5q-127,000

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  Now you have a two-step equation you can solve.
  ::现在你有一个两步方程 你可以解决。

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  $$\begin{array}{rl}400,000& =8.5q-127,000\\ 527,000& =8.5q\\ q& =62,000\end{array}$$

  
  ::400,000=8.5q-17,000,527,000=8.5qq=62,000

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  Patrick will have to sell 62,000 shirts over the course of the year.
  ::Patrick要卖62 000件衬衫

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  Discussion Questions
  ::讨论问题 讨论问题

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  1. You combined $\begin{array}{r}15q\end{array}$  and $\begin{array}{r}-6.5q\end{array}$  to get $\begin{array}{r}8q.\end{array}$  What does this $\begin{array}{r}8q\end{array}$  represent?
     ::你结合了15Q和6.5q来得到8q。这8q代表什么?
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  2. Does the answer "62,000 shirts" make sense within the context of the problem? How can you check your answer mathematically? Does it check out mathematically?
     ::答案“ 62,000 件衬衫” 在问题的背景中是否有意义? 您如何用数学方法检查您的答案? 它是否用数学方法检查?

   

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  Distribution
  ::分发分发情况

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  are seen across many professional fields. Science is another field where they regularly appear. Just think of all the equations you have seen, like $\begin{array}{r}\text{Density}=\frac{\text{Mass}}{\text{Volume}}\end{array}$   and  $\begin{array}{r}\text{Speed}=\frac{\text{Distance}}{\text{Time}}.\end{array}$  These are examples of one-step equations , but now that you know how to solve multi-step equations you will begin to see a lot more of them. One such equation is used to convert Celsius into Fahrenheit. Celsius is the temperature scale used outside of the United States and is the agreed-upon temperature scale used by all scientists.  If you see a temperature in Fahrenheit and want to convert it to Celsius, one method  is to use the formula :
  ::在许多专业领域都可以看到。 科学是它们经常出现的另一个领域。 想想你所看到的所有方程式, 比如 Density=MassVolume 和 Speed= DistanceTime 。 这些是一阶方程式的例子, 但是现在你知道如何解答多阶方程式, 你就会看到更多。 其中一种方程式用来将摄氏转换成法赫里内海特。 摄氏度是美国以外使用的温度尺度, 是所有科学家都同意使用的温度尺度。 如果您在法赫伦海特看到温度, 并且想将其转换为摄氏度, 一种方法是使用公式 :

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  $\begin{array}{r}C=\frac{5}{9}(F-32).\end{array}$
  ::C=59(F-32)。 C=59(F-32)。

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  Example
  ::示例示例示例示例

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  Use the formula above to convert 2 5° C to Fahrenheit.
  ::使用上面的公式将25°C转换为华氏25度。

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  When solving a multi-step equation with parenthesis, there are two approaches: you can distribute the number outside the parenthesis with the expression inside the parenthesis, or you can divide the equation by that number. Remember that dividing is the same as multiplying by the reciprocal .
  ::在用括号解析多步方程时,有两种方法:您可以在括号内用表达式在括号外分配数字,或者将公式除以该数字。记住,除法与对等乘法相同。

  Distribution	Dividing
  播放段落 $$\begin{array}{rl}25& =\frac{5}{9}(F-32)\\ \\ 25& =\frac{5}{9}\cdot F-\frac{5}{9}\cdot 32\\ \\ 25& =\frac{5}{9}F-\frac{160}{9}\\ \\ \text{Multiply}& \text{ by 9 to clear the fractions.}\\ 9(25)& =9\cdot \left(\frac{5}{9}F+\frac{160}{9}\right)\\ \\ 225& =5F-160\\ 385& =5F\\ 77& =F\end{array}$$ ::25=59(F-32) 25=59F-599-32 25=59=F-1609 乘以9=9(25)=9(59F+16009) 225=5F-160385=5F77=F	$$\begin{array}{rl}25& =\frac{5}{9}(F-32)\\ \\ 25\cdot \frac{9}{5}& =\frac{5}{9}(F-32)\cdot \frac{9}{5}\\ \\ 45& =F-32\\ 77& =F\end{array}$$

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   Answer: 2 5° C converts to 77° F.
  ::答复:25°C转换为77°F。

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  In a situation like this where there is a constant on one side, and there is one term with an expression in parenthesis on the other side, the division approach is easier. However, if there are more terms , the distribution approach will likely be easier.
  ::在这样的情况下,一方面有一个常数,另一方面有一个用括号表示的术语,分离法比较容易。 但是,如果有更多条件,分配法可能比较容易。

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  CK-12 PLIX Interactive: Temperature Conversion
  ::CK-12 PLIX 互动:温度转换

   

   

   

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  Combining Like Terms & Distribution
  ::将类似条件和分布合并

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  Consider this  example which involves both the and combining like terms:
  ::考虑一下这个既涉及类似术语又结合类似术语的例子:

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  Example
  ::示例示例示例示例

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  $$\begin{array}{r}2y-5(y+1)=\text{-}2\end{array}$$

  
  ::2y-5(y+1)=-2

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  You  can’t combine like terms if one term is in parenthesis and one term is outside of those parentheses , y ou must first eliminate the parenthesis. Dividing every term by -5 would not be convenient in this case because the terms $\begin{array}{r}2y\end{array}$  and $\begin{array}{r}\text{-}2\end{array}$  would not result in whole numbers.  You will distribute the -5 to each term in the parenthesis.
  ::如果括号中有一个词名,而括号中有一个词名不在括号内,则无法将类似术语合并。首先必须删除括号。在这种情况下,将每个词的括号划为 - 5并不方便,因为二字和二字词不会产生整数。您将把 - 5 词分配给括号中的每个词名。

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  $$\begin{array}{rl}2y-5(y+1)& =\text{-}2\\ 2y-5y-5& =\text{-}2\\ \text{-}3y-5& =\text{-}2\\ \text{-}3y& =3\\ y& =\text{-}1\end{array}$$

  
  ::2y-5(y+1)=-2y-5y-5y-5=-2-3y-5=-2-3y-5=-2-3y=-2-3y=3y=-1

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  Use the distributive property to solve the equation in the interactive below.  Press refresh to try again with a different set of numbers.
  ::使用分配属性解析下面交互的方程式中的方程式。按下刷新以不同的数字组重试 。

  

  INTERACTIVE

  Linear Equation Distribution

  

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  Follow Flexi's steps through the problem.
  ::跟着Flexi的步子 解决问题。

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     Summary
  ::摘要

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  A multi-step equation is an equation that requires more than two steps to solve for the variable.
  ::多步方程是一个需要两个以上步骤才能解决变量的方程。

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  Some steps may include:
  ::一些步骤可包括:

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  • Distribution, for example:  $\begin{array}{r}2(x+10)+3=2x+20+3\end{array}$  
    ::分布,例如:2(x+10)+3=2x+20+3
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  • Combining like terms, for example:  $\begin{array}{r}2x+20+3=2x+23\end{array}$  
    ::组合类似术语, 例如: 2x+20+3=2x+23