5.1 线性等式理解系统-interactive

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  线性等式理解系统

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  Going to College
  ::上学院

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  In this chapter,  you'll explore what systems of equations are and how they are used. The professional lens through which we view systems of equations is that of an analyst. An analyst is someone who examines data and real-world mathematical relationships to find information. This information is then used to make the decisions which shape our world. Analysts help people and businesses make smart financial decisions, help the FBI find criminals, help sports teams evaluate players, and much more. Analysts generally use the following step by step process to help make decisions:
  ::在本章中, 您将探索方程系统是什么以及它们是如何使用的。 我们查看方程系统的专业透镜是分析师的。 分析师是分析数据和真实世界的数学关系以寻找信息的人。 然后, 这些信息被用来做出决定决定塑造我们的世界。 分析师帮助人们和企业做出明智的金融决策, 帮助联邦调查局找到罪犯, 帮助体育队评估球员, 以及更多。 分析师通常使用以下一步步的流程来帮助做出决策 :

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  1. Identify the Problem
     ::找出问题
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  2. Gather Data and Research
     ::收集数据和研究
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  3. Analyze the Data
     ::分析数据
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  4. Take Action
     ::采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动采取行动 采取行动采取行动 采取行动采取行动采取行动 采取行动采取行动采取行动采取行动采取行动采取行动 采取行动采取行动采取行动采取行动采取行动采取行动 采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动

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  You  will be referring back to these steps throughout the chapter as you examine a wide range of real-world situations. The first situation  you will take a  look at regards one of the first major decisions you will have to make: going to college.  You will take an analytical approach to the question, “What are the financial benefits of going to college?”
  ::在整个章节中,你将回顾这些步骤,研究一系列广泛的现实世界情况。第一种情况是,你将审视你必须作出的第一批重大决定之一:上大学。你将分析一下这个问题,即“上大学有什么财政好处?”

   

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  Comparing Rates In Tables
  ::表中的比较率

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  The question above can be answered using a system of equations . A system of equations is two or more equations with the same set of variables. This means that the equations relate to each other.  You can compare the values for the variables of both equations to determine information about the situation. The first situation relates to examining flow rates. Plumbers and engineers commonly face this problem rather than analysts, but it is a great tangible example to help one understand the concept. Engineers and plumbers need to have a strong understanding of flow rates when designing drainage systems. Fluids need to be able to leave the system faster than they enter to prevent back-up and overflow. The ability to measure the rate at which a fluid enters and leaves a system is vital to the performance of the system.
  ::上述问题可以通过一个方程式系统解答。一个方程式系统是两个或两个以上的方程式,具有相同的变量组。这意味着方程式是相互联系的。您可以比较两个方程式变量的数值,以决定有关形势的信息。第一种情况是检查流量率。管道工和工程师通常面临这一问题,而不是分析师,但这是一个帮助人们理解这个概念的非常具体的例子。工程师和水管工在设计排水系统时需要深入了解流量率。流体需要能够比输入系统更快离开系统,以防止备份和溢出。测量液体进入和离开系统的速度对于系统的运作至关重要。

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  Example
  ::示例示例示例示例

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  There are two water bottles that each hold 16.9 fluid ounces. Bottle A is slowly being filled with water at an average rate of 0.8 fluid ounces per second. Bottle B is full and has a small hole poked in it and is leaking water at an average rate of 0.5 fluid ounces per second. If bottle B starts to drain at the same time that bottle A begins to be filled, at what time will bottle A and bottle B have the same amount of water?
  ::瓶子A 正在缓慢地以每秒0.8液盎司的平均速率向水中充水。 瓶子B 满了,并有一个小孔插在瓶子里,每秒平均0.5液盎司的漏水速率。 如果瓶子B在瓶子A开始装满时开始排水,瓶子A和瓶B何时能有同样的水量?

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  Use the interactive below to visualize the example. 
  ::使用下面的交互式图像来直观示例 。

   

   

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  Use the interactive to  make a table of  how much water is in each water bottle after  $\begin{array}{r}x\end{array}$   seconds.
  ::使用交互方式绘制一个表格,显示每个水瓶在x秒后有多少水。

   

   

   

   

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  The point at which the two rates produce equal amounts is what we call the solution to the system of equations .
  ::这两种汇率产生相同数额的点是我们称之为方程体系解决办法的点。

   

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  Comparing Rates In Graphs
  ::图中比较率

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  In the example above , you used a table to solve a system of equations. However, what if the answer was 100 seconds? Clearly it would be inefficient to write out a 100 entry table and search for the moment at which the two amounts of water were equal. Additionally, if the two amounts of water are equal at a time that is not a whole number , your table would not show it. Another way to visualize and solve a system of equations is by graphing the linear equations which represent the relationship.
  ::在以上的例子中, 您使用一张表格来解决一个方程式系统。 但是, 如果答案是 100 秒, 那么它会如何 ? 显然, 写出一个 100 个输入表, 并搜索两个水量相等的时间是无效的 。 此外, 如果两个水量相等于一个不是整数的时间, 您的表格将不会显示它 。 另一种可视化和解析一个方程式系统的方法是绘制代表关系的线性方程式图 。

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  To graph a linear equation , you will  need two variables:  $\begin{array}{r}x\end{array}$  and  $\begin{array}{r}y.\end{array}$  What would these represent in the context of the problem?
  ::要绘制线性方程图,您需要两个变量: x 和 y。在问题的背景中,这些变量代表什么?

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  The $\begin{array}{r}x\text{-}\end{array}$ axis, or  $\begin{array}{r}x\end{array}$  value will represent by time in seconds, and the $\begin{array}{r}y\text{-}\end{array}$ axis, or  $\begin{array}{r}y\end{array}$  value will represent the amount of water in fluid ounces.
  ::x 轴, 或 x 值将按时间以秒表示, Y 轴, 或 y 值将代表液体盎司中的水量 。

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  What equation can be used to the amount of water,  $\begin{array}{r}y,\end{array}$  in bottle A, at  $\begin{array}{r}x\end{array}$  seconds?
  ::用什么方程式来计算水量, y,瓶A,x秒?

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  B ottle A begins with no water and fills at an average rate of 0.8 fluid ounces per second which can be modeled by the equation  $\begin{array}{r}y=0.8x.\end{array}$
  ::瓶装A从没有水开始,以每秒0.8盎司的平均速率填充水,以y=0.8x等式模拟。

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  What equation can be used to the amount of water,  $\begin{array}{r}y,\end{array}$  in bottle B , at  $\begin{array}{r}x\end{array}$  seconds?
  ::用什么方程式来计算水量,y,瓶B,x秒?

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  B ottle  B begins with 16.9 ounces of water and empties at an average rate of 0.5 fluid ounces per second which can be modeled by the equation  $\begin{array}{r}y=16.9-0.5x.\end{array}$
  ::瓶装B以16.9盎司的水和水槽开始,平均速度为每秒0.5立方体,以y=16.9-0.5x为模型。

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  Use the interactive below to graph the linear equations. 
  ::使用下面的交互式方程式绘制线性方程图。

   

   

   

   

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  The solution to a system of equations is also called the point of intersection . To intersect means to meet or cross paths. The equations cross paths at the ordered pair  $\begin{array}{r}(x,y)\end{array}$  that makes both equations true. You  can use this knowledge to easily identify the solution to a system of equations when the equations are graphed.
  ::方程式系统的解决方案也被称为交叉点。 交叉方式是指相交或交叉路径。 使两个方程式都真实的定对( x,y)的方程式交叉路径。 您可以使用这种知识来很容易地识别方程式系统中的解决方案, 当方程式被图形化时 。

   

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  G oing to College Continued
  ::继续上大学

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  The problem in the introduction  asked , “What are the benefits of going to college?” Perform an analysis of this question using the steps analysts use as discussed in the introduction.
  ::在导言中提出的问题是,“上大学有什么好处?” 利用导言中讨论的步骤分析师使用的方法分析这一问题。

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  Identify the Problem
  ::找出问题

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  College is expensive and a big commitment.  You want to know what the financial benefits are before  investing such a large amount of money.
  ::大学是昂贵的,也是一笔巨大的承诺。你想知道在投资如此巨额的资金之前,经济效益是什么。

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  Gather Data and Research
  ::收集数据和研究

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  There are many different types of colleges.  You will look at private universities in this example which are generally on the more expensive side. There are close to 2,000 private universities in the United States, and examples of private universities include Stanford, NYU, Yale, and MIT. On average, a private university costs about $32,000 per year which is about $128,000 for a four-year program. The average college graduate earns approximately $62,000 per year. In contrast, a high school graduate earns approximately $34,700 per year and has a 4-year head start on earning money.
  ::有多种不同类型的学院。您将在此例子中看到私人大学,这些大学一般比较昂贵。美国有近2,000所私立大学,私立大学的例子包括斯坦福大学、纽约大学、耶鲁大学和麻省理工学院。平均而言,私立大学每年花费约32,000美元,四年制课程约为128,000美元,大学毕业生平均每年收入约62,000美元。相比之下,高中毕业生每年收入约34,700美元,从赚取收入开始,为期4年。

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  Analyze the Data
  ::分析数据

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  Either create a table or graph the linear equations to determine when the college graduate and the high school graduate will have earned the same amount of money. S tart the table or graph at year 0 which is the moment  a college student graduates. Remember that at this point the college student is $128,000 in debt and the high school student has made $138,800.
  ::要么创建表格,要么绘制线性方程图,以确定大学毕业生和高中毕业生何时能挣得相同数额的钱。 从零年级开始,也就是大学生毕业生的那一时刻。 记住,此时该大学生债务为128 000美元,高中学生债务为138 800美元。

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  Use the interactive below to analyze the situation presented above.
  ::利用下文的交互分析分析上述情况。

   

   

   

   

   

   

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  Take Action
  ::采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动 采取行动采取行动 采取行动采取行动 采取行动采取行动采取行动 采取行动采取行动采取行动采取行动采取行动采取行动 采取行动采取行动采取行动采取行动采取行动采取行动 采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动

  播放段落
  1. Now  you can answer the question “What are the financial benefits of going to college?"
     ::现在,你可以回答一个问题:“上大学的经济利益是什么?”
  播放段落
  2. What happens after the high school graduate and college graduate have earned the same amount of money?
     ::高中毕业生和大学毕业生赚到相同数额的钱之后会怎么样?

  Summary

  播放段落 The solution to a system of equations is the point at which the two equations have the same input value and the same output. ::方程式系统的解决方案是两个方程式具有相同输入值和相同输出值的点。 播放段落 The solution on the graph of a system of equations is the point where the two lines intersect. ::方程系统图上的解决方案是两条线相交的点。