标准表格中的线性公式

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What is Standard Form?
::什么是标准表格?

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You know that to graph a linear equation that is written in slope-intercept form you can plot the $\begin{array}{r}y\text{-}\end{array}$ intercept and then use the to find another point along the line quickly. This works so effectively because you can quickly get two points which are the minimum number of points needed to graph a line. It is possible for linear equations to take forms other than slope-intercept form and in this section, you will learn how to graph an equation when this happens.
::您知道, 要绘制以斜坡截取形式写成的线性方程式, 您可以绘制 y 截取, 然后快速地在线上找到另一个点 。 此功能非常有效, 因为您可以快速获得两个点, 这是绘制线性方程式所需的最小点数 。 线性方程式可以采用斜坡截取形式以外的形式, 而在本节中, 您将学习如何在此情况下绘制一个方程式 。

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The form which you will be exploring in this section is standard form . Standard form is a method for writing linear equations in the format $\begin{array}{r}Ax+By=C.\end{array}$  The variables $\begin{array}{r}A,\end{array}$   $\begin{array}{r}B,\end{array}$  and $\begin{array}{r}C\end{array}$  are used to identify coefficients and constants but do not have the same graphing significance the slope and ^{$\begin{array}{r}y\text{-}\end{array}$} intercept. A simple way to think of standard form is when both variables are on the same side of the equal sign with the remaining constant on the other side. This form is used to write an equation which combines two related rates. This can be seen in geometric contexts, in business contexts and many more.  Later in this section, you will look at how hospitals use linear equations in standard form to solve staffing problems.
::在本节中您要探索的窗体是标准格式。 标准窗体是一种以 Ax+By=C 格式书写线性方程式的方法。 变量A、 B和 C 用于确定系数和常数, 但没有同样的图形意义, 即斜度和 y- intercept 。 标准窗体的简单思维方式是当两个变量都位于等号的同一侧面, 与另一侧的剩余常数相同时。 此窗体用于写入一个将两个相关速率结合起来的方程式。 这可以在几何环境中、 在商业环境中和更多情况下看到。 在此节的后面, 您将查看医院如何使用标准形式的线性方程式来解决人员配置问题 。

 

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Graph By Plotting Points
::按绘图点绘制的图表

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One of the simplest methods for graphing equations that are not in standard form is to plot points. This was the strategy explored in . While the idea has not changed, the approach will be slightly different.
::不以标准格式绘制方程式的最简单方法之一是绘制点。 这是在 . 中探讨的战略。 虽然这个想法没有改变, 但方法会略有不同 。

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Example
::示例示例示例示例

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Nectaria owns a farm and is building a rectangular pen for her five sheep. She wants to use the 60 feet of fencing that she already has. Graph the linear relationship between the length and the width of the pen.
::Nectaria拥有一个农场,正在为其五只羊建造一条长方形笔,她想使用她已有的60英尺围栏。绘制笔的长度和宽度之间的线性关系图。

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To graph this, you must begin with an equation. However, there is no increasing or decreasing rate that jumps out. This situation has two related rates: length and width. To write an equation which will state the linear relationship between length and width, you will need to use the perimeter formula : $\begin{array}{r}P=2l+2w.\end{array}$  You can substitute 60 in for $\begin{array}{r}P\end{array}$  because the perimeter of the pen must be 60 ft. You can also use $\begin{array}{r}x\end{array}$  to represent the length and $\begin{array}{r}y\end{array}$  to represent the width to use letters that you may be more comfortable graphing. This gives you your final equation:
::要绘制此公式, 您必须从方程开始。 但是, 跳出的速度不会增加或减少。 这一情况有两个相关速度: 长度和宽度。 要写出一个公式来说明长度和宽度之间的线性关系, 您需要使用周边公式: P=2l+2w。 您可以用60 英寸代替P, 因为笔的周长必须是 60英尺。 您也可以用 x 表示长度, y 表示宽度, 以便使用您可能更舒适的图形化字母。 这给了您最后的公式 :

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::60=2x+2y

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The solutions to a linear equation represent the combinations of values for $\begin{array}{r}x\end{array}$  and $\begin{array}{r}y\end{array}$  that make the equation true. In this case, the solutions to this equation represent the combinations of length and width that will give us 60 feet. To find a pair of length and width values which make the equation true you must choose one value and solve for the corresponding value. Let your first $\begin{array}{r}x\text{-}\end{array}$ value be 1. To find the corresponding $\begin{array}{r}y\text{-}\end{array}$ value, you must substitute 1 in for $\begin{array}{r}x\end{array}$  and solve for  $\begin{array}{r}y.\end{array}$
::线性方程式的解决方案代表 x 和 y 的值组合, 使方程式成为真实。 在这种情况下, 此方程式的解决方案代表长度和宽度的组合, 给我们提供60英尺。 要找到使方程式成为真实的长度和宽度的一对数值, 您必须选择一个值, 并解决相应的值。 将第一个 x 值定为 1 。 要找到相应的 y 值, 您必须替换 x 中的 1, 并解决 y 。

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::60=2(1)+2y60=2+2y58=2y29=y

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This means that with 60 feet of fencing, Nectaria can make a pen that is 1 feet by 29 feet. However, these dimensions would make it difficult for the sheep to move around. Use the interactive plot more points to see what all of the possible options are for the length and width of the sheep pen.
::这意味着,用60英尺的栅栏,Nectaria可以制造一英尺乘29英尺的钢笔,然而,这些尺寸会使绵羊难以移动。 使用互动图段,用更多的点子来观察绵羊钢笔的长度和宽度,所有可能的选项是什么。

 

 

 

 

 

 

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Discussion Questions
::讨论问题 讨论问题

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1. What are some combinations of length and width that are reasonable? What are some combinations of length and width that are unreasonable?
   ::哪些长度和宽度的组合是合理的?哪些长度和宽度的组合是不合理的?哪些长度和宽度的组合是不合理的?
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2. How many possible combinations are shown on the line you graphed? Are they all possible?
   ::您绘制的线上显示了多少可能的组合? 它们都有可能吗 ?

 

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Graph By Plotting the Intercepts
::绘制截取图

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Another real-world application for linear equations written in standard form is to model the potential distributions of items of differing price or size within a limit.
::另一种以标准格式书写线性方程式的现实世界应用程序是模拟价格或大小不同的物品在一定限度内的潜在分布。

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Example
::示例示例示例示例

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Cameron goes to the store with $18 to buy chips and sodas for a party he is having. A bag of chips costs $3 and a soda costs $2. Graph the linear equation which shows all possible combinations of the number of bags of chips and the number of sodas that Cameron can buy.
::卡梅伦带着18美元去商店买薯片和苏打水,他正在开派对。一袋薯片要3美元,苏打水要2美元。用线性方程图显示所有可能的芯片袋数和卡梅伦可以买的苏打水的组合。

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First, you will need to write an equation to model this situation. Our two variables are the number of bags of chips, represented by $\begin{array}{r}x,\end{array}$  and the number of sodas, represented by $\begin{array}{r}y.\end{array}$  The cost of $\begin{array}{r}x\end{array}$  bags of chips is $\begin{array}{r}3x\end{array}$  dollars and the cost of $\begin{array}{r}y\end{array}$  sodas is $\begin{array}{r}2y\end{array}$  dollars. If you add these together, you will get $18. This gives you the following equation:
::首先,您需要写一个方程式来模拟这种情况。 我们的两个变量是: 以x为代表的芯片袋数; 以y为代表的苏打水数。 x袋芯片的费用是 3x美元, y苏打水的费用是 2 y 美元。 如果把这些加在一起, 您将得到 18 美元。 这样可以得出以下的方程式 :

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::3x+2y=18

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Another method for graphing linear equations in standard form is to plot the intercepts . Recall from previous sections in this chapter that the $\begin{array}{r}y-\end{array}$ intercept is located at the point $\begin{array}{r}(0,b)\end{array}$  and the $\begin{array}{r}x\text{-}\end{array}$ intercept is located at the point $\begin{array}{r}(c,0).\end{array}$  You can take advantage of this and substitute 0 in for $\begin{array}{r}x\end{array}$  to find the $\begin{array}{r}y\text{-}\end{array}$ coordinate of the $\begin{array}{r}y\text{-}\end{array}$ intercept. You can also substitute 0 in for $\begin{array}{r}y\end{array}$  to find the $\begin{array}{r}x\text{-}\end{array}$ coordinate of the $\begin{array}{r}x\text{-}\end{array}$ intercept. Even though two points are enough to graph a linear equation, you will choose a random $\begin{array}{r}x\text{-}\end{array}$ value for the third point to check your answers. If all three points line up, you graphed the relationship correctly. Use the interactive below to graph the equation  $\begin{array}{r}3x+2y=18.\end{array}$
::以标准格式绘制线性方程式的另一种方法就是绘制截取的图。 回顾本章前几节, y- 截取位于点( 0, b) 和 x 截取位于点( c, 0) 。 您可以利用这个方法, 并用 0 替换 x 来查找 y 截取的 Y 坐标 。 您也可以用 0 替代 y 来查找 x 截取的 x 坐标 。 即使两个点足以绘制线性方程式, 您也可以为第三个点选择一个随机的 x 值来检查您的答案 。 如果所有三个点都向上, 您可以正确绘制关系图示 。 使用下面的交互式图解 3x+2y=18 。

 

 

 

 

 

 

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Discussion Questions
::讨论问题 讨论问题

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1. Are there any solutions along the line that are not possible combinations given the context?
   ::鉴于背景情况,是否有任何不可能结合的解决办法?
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2. Are there any solutions along the line that are possible solutions but wouldn’t make sense given the context?
   ::是否有任何解决方案可以解决, 但从背景来看却不合理?

 

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Graph By Converting to Slope-Intercept Form
::通过转换为斜坡截取窗体的图形

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The final method you will explore to graph a linear equation that isn’t in slope-intercept form is to convert it to slope-intercept form. This can be accomplished by solving for $\begin{array}{r}y\end{array}$  the same way that you solved a two-step equation :
::您将探索的绘制线性方程式的最后方法不是以斜度界面形式绘制的线性方程式是将其转换成斜度界面。 可以通过解决y的方式实现这一点, 与解决两步方程式的方法相同 :

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1. Move any terms that are connected to $\begin{array}{r}y\end{array}$  by addition or subtraction .
   ::移动任何与 y 相连接的用增减方式连接的术语。
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2. Divide each term by the coefficient of  $\begin{array}{r}y.\end{array}$  
   ::将每一术语除以y系数。

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One place you see standard form is in nurse staffing where there needs to be a specific ratio of nurses to patients.  
::你看到标准表格的一个地方是护士的人员配置,需要有一个护士与病人的具体比例。

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Example
::示例示例示例示例

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In a hospital, there is a floor for patients who are undergoing and recovering from surgery. There needs to be a nurse for each patient in an operating room and a nurse for every 2 patients who are recovering from surgery. There are currently 12 nurses in the hospital. Let $\begin{array}{r}x\end{array}$  represent the number of patients in surgery and y represent the number of patients in recovery. You can model the possible number of patients with the equation $\begin{array}{r}x+2y=12.\end{array}$  Graph the possible numbers of patients that these nurses can support.
::在医院里,有一层楼供正在接受手术和从手术中恢复的病人使用,每个病人在手术室需要有一名护士,每2名正在从手术中恢复的病人需要有一名护士,目前医院里有12名护士。让x代表手术中的病人人数,y代表正在康复的病人人数。你可以用公式x+2y=12来模拟可能的病人人数。请列出这些护士能够支持的病人的可能人数。

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1. Move any terms that are connected to $\begin{array}{r}y\end{array}$  by addition and subtraction.
   ::移动与 y 相连接的词句,加上和减。

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To isolate the $\begin{array}{r}y\end{array}$  term, you must subtract $\begin{array}{r}x\end{array}$  to both sides. Remember that you cannot combine $\begin{array}{r}x\end{array}$  and 12 because they are not like terms .
::要孤立 y 术语, 您必须将 x 减到两边 。 请记住, 您不能将 x 和 12 合并, 因为它们和 12 不一样 。

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$\begin{array}{rl}& x+2y=12\\ & \underset{\_}{-x-x}\\ & 2y=12-x\end{array}$
::x+2y=12-x-x_2y=12-x

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2. Divide each term by the coefficient of  $\begin{array}{r}y.\end{array}$  
   ::将每一术语除以y系数。

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The coefficient of the variable $\begin{array}{r}y\end{array}$  is 2 so divide each term by 2.
::变量y的系数为 2, 所以每个术语除以 2。

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$\begin{array}{rl}& 2y=12-x\\ \\ & \frac{2}{2}y=\frac{(12-x)}{2}\\ \\ & \frac{2}{2}y=\frac{12}{2}-\frac{x}{2}\\ \\ & y=6-\frac{1}{2}x\end{array}$
::2y=12-x22y=(12-x)2222y=122-x2y=6-12x

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Since the coefficient of $\begin{array}{r}x\end{array}$  represents the slope, it will be easier to write it as a fraction . Now you can easily identify the rise and the run .
::由于 x 系数代表斜坡, 将其写成一个分数比较容易。 现在您可以很容易地识别上升和运行 。

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Use the interactive below to explore the possible combinations of each type of patient.  
::使用下面的交互方式来探讨每一种病人的可能组合。

 

 

 

 

 

 

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Discussion Questions
::讨论问题 讨论问题

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1. Are there any solutions along the line that are not possible combinations given the context?
   ::鉴于背景情况,是否有任何不可能结合的解决办法?
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2. One of the points on the line is (2, 5) What does this point represent?
   ::这条线上的一个点是(2,5),这个点代表什么?
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3. Are there any solutions along the line that are possible solutions but wouldn’t make sense given the context?
   ::是否有任何解决方案可以解决, 但从背景来看却不合理?