10.2 两个等同和两个未知的系统

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• 选择节两个等量和两个未知星系

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  两个等量和两个未知星系

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  Introduction
  ::导言

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  The cost of two cell phone plans can be written as a system of equations based on the number of minutes used and the base monthly rate . 
  ::两个移动电话计划的费用可以写成一个方程系统,根据所用分钟数和每月基本费率计算。

  

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  As a consumer, you would find it useful to know when the two plans cost the same, and when one plan is less expensive. 
  ::作为消费者,你应该知道,这两个计划的成本是相同的,一个计划的成本是较低的。

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  Plan A costs $40 per month plus $0.10 for each minute of talk time. 
  ::A计划每月费用40美元,每发言一分钟加0.10美元。

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  Plan B costs $25 per month plus $0.50 for each minute of talk time. 
  ::B计划每月费用为25美元,每发言一分钟加0.50美元。

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  Plan B has a lower starting cost, but it costs more per minute. Therefore , it may not be the most cost-effective  plan for someone who likes to spend a lot of time on the phone.
  ::B计划起点成本较低,但每分钟成本更高。 因此,对于喜欢在电话上花很多时间的人来说,它可能不是最具成本效益的计划。

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  When do the two plans cost the same amount?
  ::这两项计划何时费用相同?

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  Solving Systems of Equations with Two Unknowns
  ::两种未知物的平方溶解系统

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  There are many ways to solve a system of equations, including substitution, elimination, and graphing. Here we will focus on solving using elimination, because the knowledge and skills we use with this method will transfer directly to our work with matrices. When we solve systems of linear equations, there are three solution options:
  ::有很多方法可以解决方程式系统,包括替代、消除和图形化。在这里,我们将集中解决消除问题,因为我们使用这种方法的知识和技能将直接转移到我们使用矩阵的工作上。当我们解决线性方程式系统时,有三个解决方案:

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  1. If a system is  independent , then the linear equations intersect at one point, so there is one solution to the system. 
     ::如果一个系统是独立的,那么线性方程式就在一个点交叉,因此系统有一个解决办法。
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  2. If a  system is  dependent , then the linear equations intersect at every point on the lines, so  there are infinitely many  solutions to the system. 
     ::如果一个系统是依赖的,那么线性方程式就会在线条的每个点上交叉,所以这个系统有无限多的解决方案。
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  3. If a system is  inconsistent , then the linear equations are parallel , so there are no  solutions to the system. 
     ::如果一个系统不一致,那么线性方程式是平行的,因此对系统就没有解决办法。

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  When using elimination to solve a system, first  count the number of variables that are missing and the number of equations. The number of variables needs to be the same or fewer than the number of equations. For instance, a system with two equations and two variables can be solved, but one equation with two variables cannot. 
  ::当使用删除来解析一个系统时, 首先计算缺失的变量数和方程数。 变量数需要与方程数相同或更少。 例如, 一个包含两个方程和两个变量的系统可以解答, 但一个带有两个变量的方程无法解答 。

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  Here's the procedure for solving a system using the elimination method :
  ::使用消除方法解决系统的程序如下:

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  • Step 1 : Write both equations with two variables in standard form , $\begin{array}{r}Ax+By=C\end{array}$ .  This form helps to align the variables. 
    ::第1步:用标准格式的两个变量Ax+By=C写两个方程式。这个方程式有助于对齐变量。
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  • Step 2 : Determine which variable you want to eliminate.
    ::第2步:确定要删除的变量。
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  • Step 3 : Scale each equation as necessary by multiplying through by constants.
    ::第3步:通过乘以常数,根据需要对每个方程式进行比例。
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  • Step 4 : Add the equations together.
    ::第4步:将方程加在一起。
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  • Step 5 :
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    • If the system is independent, solve for the variable. Then s ubstitute the result in to one of the linear equations to  determine the value for the second variable.  
      ::如果系统是独立的, 请解决变量。 然后将结果替换为线性方程之一, 以确定第二个变量的值 。
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    • If the system is dependent, Step 4 results in  $\begin{array}{r}0=0.\end{array}$  
      ::如果系统有依赖性,第4步的结果为0=0。
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    • If the system is inconsistent , Step 4 results in  $\begin{array}{r}0=k,\end{array}$  where  $\begin{array}{r}k\end{array}$  is a nonzero number.
      ::如果系统不一致,第4步的结果为0=k,K是非零数。
    
    ::第5步:如果系统是独立的,请解决变量。然后将结果替换成线性方程式之一,以决定第二个变量的值。如果系统是独立的,则第4步以0=0计算。如果系统不一致,则第4步以0=k计算,K是非零数。

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  Here is a system of two equations and two variables in standard form :  $\begin{array}{r}5x+12y=72\end{array}$ and $\begin{array}{r}3x-2y=18\end{array}$ .  Notice there is an  $\begin{array}{r}x\end{array}$ column and a  $\begin{array}{r}y\end{array}$ column on the lefthand side, and a constant column on the righthand side. If not, rewrite the equations as shown. Also notice that if you add the system as written, no variable will be eliminated. 
  ::标准格式为 5x+12y= 72 和 3x-2y= 18 。 注意左侧有 x 列和 y 列,右侧有 常数列。 否则, 请重写所显示的方程 。 请注意, 如果您将系统添加成文字, 将不会删除变量 。

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  Equation 1:   $\begin{array}{r}5x+12y=72\end{array}$
  ::等式1: 5x+12y=72

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  Equation 2:   $\begin{array}{r}3x-2y=18\end{array}$
  ::等式2: 3x-2y=18

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  Strategically choose to eliminate  $\begin{array}{r}y\end{array}$ by scaling the second equation by 6, so that the coefficient of  $\begin{array}{r}y\end{array}$ will match at 12 and -12.
  ::从战略上选择通过将第二个方程的乘以6来消除y,使y的系数与12和12相匹配。

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  $$\begin{array}{rl}5x+12y& =72\\ 18x-12y& =108\end{array}$$

  
  ::5x+12y=7218x-12y=108

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  Add the two equations:
  ::添加两个方程式:

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  $$\begin{array}{rl}23x& =180\\ x& =\frac{180}{23}\end{array}$$

  
  ::23x=180x=18023

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  The value for  $\begin{array}{r}x\end{array}$ could be substituted into either of the original equations, and the result could be solved for $\begin{array}{r}y\end{array}$ ; however, since the value is a fraction , it may be easier to repeat the elimination process to solve for $\begin{array}{r}y\end{array}$ . This time you will take the first two equations and eliminate  $\begin{array}{r}x\end{array}$ by making the coefficients of  $\begin{array}{r}x\end{array}$ to be 15 and -15. Scale the first equation by a factor of 3, and scale the second equation by a factor of -5.
  ::x 的值可以替换为原始方程中的任何一个,结果可以为y解决;然而,由于该值是一个分数,因此可能更容易重复y的消除过程。 这次你将采用前两个方程,通过将x的系数定为15和-15来消除x。将第一个方程乘以3,将第二个方程乘以5,然后将第二个方程乘以5,将第一个方程乘以3。

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  Equation 1:  $\begin{array}{r}15x+36y=216\end{array}$
  ::等式1: 15x+36y=216

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  Equation 2:  $\begin{array}{r}-15x+10y=-90\end{array}$
  ::等式2: - 15x+10y90

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  Adding the two equations:
  ::添加两个方程式:

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  $$\begin{array}{rl}0x+46y& =126\\ y=\frac{126}{46}& =\frac{63}{23}\end{array}$$

  
  ::0x+46y=126y=12646=6323

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  The point $\begin{array}{r}\left(\frac{180}{23},\frac{63}{23}\right)\end{array}$  is where these two lines intersect.   
  ::要点(18023,6323)是这两条线交错之处。

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  The following video has further examples  demonstrating how to solve a system of equations using the elimination method:
  ::以下视频还有更多例子,说明如何使用消除方法解决方程式系统:

   

   

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  Examples
  ::实例

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  Example 1
  ::例1

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  Recall the problem  from the Introduction in which y ou were asked w hen the two cell phone plans cost the same amount.
  ::回顾导言中的问题,即当两个手机计划费用相同时,有人问过你。

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  Plan A costs $40 per month plus $0.10 for each minute of talk time. 
  ::A计划每月费用40美元,每发言一分钟加0.10美元。

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  Plan B costs $25 per month plus $0.50 for each minute of talk time. 
  ::B计划每月费用为25美元,每发言一分钟加0.50美元。

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  Solution:
  ::解决方案 :

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  To find out when the two plans cost the same, represent each plan with an equation and solve the system of equations. Let  $\begin{array}{r}y\end{array}$ represent cost, and  $\begin{array}{r}x\end{array}$ represent number of minutes.
  ::要知道这两个计划的成本何时相同, 请用方程代表每个计划, 并解析方程系统。 让y 代表成本, x 代表分钟数 。

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  $$\begin{array}{rl}y& =0.10x+40\\ y& =0.50x+25\end{array}$$

  
  ::y=0.10x+40y=0.50x+25

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  First put these equations in standard form.
  ::首先把这些方程 以标准的形式。

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  $$\begin{array}{rl}x-10y& =-400\\ x-2y& =-50\end{array}$$

  
  ::x- 10y- 4000x- 2y- 50

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  Then scale the second equation by -1, add the equations together, and solve for $\begin{array}{r}y\end{array}$ .
  ::然后将第二个方程缩放到 - 1, 将方程加在一起, 并为 y 解决 。

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  $$\begin{array}{rl}-8y& =-350\\ y& =43.75\end{array}$$

  
  ::-8 -350y=43.75

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  To solve for $\begin{array}{r}x\end{array}$ , scale the second equation by -5, add the equations together, and solve for $\begin{array}{r}x\end{array}$ .
  ::要解析 x, 将第二个方程缩放为 5 , 将方程加在一起, 并解析 x 。

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  $$\begin{array}{rl}-4x& =-150\\ x& =37.5\end{array}$$

  
  ::-4x=150x=37.5

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  The equivalent costs of plan A and plan B will occur at 37.5 minutes of talk time with a cost of $43.75.  
  ::计划A和计划B的同等费用将以37.5分钟的谈话时间计算,费用为43.75美元。

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  Example 2
  ::例2

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  Solve the following system of equations:
  ::解决以下方程式系统:

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  $$\begin{array}{rl}6x-7y& =8\\ 15x-14y& =21\end{array}$$

  
  ::6-7y=815x-14y=21

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  Solution:
  ::解决方案 :

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  Scaling the first equation by -2 will allow the  $\begin{array}{r}y\end{array}$ term to be eliminated when the equations are summed. 
  ::以 - 2 缩放第一个等式, 当等式总和时可以删除 Y 词 。

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  $$\begin{array}{rl}-12x+14y& =-16\\ 15x-14y& =21\end{array}$$

  
  ::- 12x+14y1615x-14y=21

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  The sum is
  ::金额是

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  $$\begin{array}{rl}3x& =5\\ x& =\frac{5}{3}.\end{array}$$

  
  ::3x=5x=53。

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  Substitute  $\begin{array}{r}x\end{array}$ into the first equation to solve for $\begin{array}{r}y\end{array}$ .
  ::将 x 替换到第一个要为y 解答的方程式中 。

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  $$\begin{array}{rl}6\cdot \frac{5}{3}-7y& =8\\ 10-7y& =8\\ -7y& =-2\\ y& =\frac{2}{7}\end{array}$$

  
  ::653-7y=810-7y=8-7y=8-7y2y=27

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  The point $\begin{array}{r}\left(\frac{5}{3},\frac{2}{7}\right)\end{array}$  is where these two lines intersect.
  ::点(53,27)是这两条线交错之处。

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  Example 3
  ::例3

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  Solve the following system using elimination:
  ::使用消除方法解决以下系统:

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  $$\begin{array}{rl}5x-y& =22\\ -2x+7y& =19\end{array}$$

  
  ::5-y=22-2x+7y=19

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  Solution:
  ::解决方案 :

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  Start by scaling the first equation by 7, and notice that the  $\begin{array}{r}y\end{array}$ coefficient will immediately be eliminated when the equations are summed.
  ::开始将第一个方程缩放为 7 , 并注意当对等方程进行总和时, y 系数将立即取消 。

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  $$\begin{array}{rl}35x-7y& =154\\ -2x+7y& =19\end{array}$$

  
  ::35-7y=154-2x+7y=19

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  Add; solve for $\begin{array}{r}x=\frac{173}{33}\end{array}$ .  Instead of substituting, practice eliminating  $\begin{array}{r}x\end{array}$ by scaling the first equation by 2 and the second equation by 5.
  ::添加; 解析 x= 173333。 做法不是替换,而是通过将第一个方程式缩放2 和第二个方程式缩放5 来删除 x。

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  $$\begin{array}{rl}10x-2y& =44\\ -10x+35y& =95\end{array}$$

  
  ::10--2y=44-10x+35y=95

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  Add; solve for $\begin{array}{r}y\end{array}$ . 
  ::添加; y 的解析。

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  Final Answer:  $\begin{array}{r}\left(\frac{173}{33},\frac{139}{33}\right)\end{array}$
  ::最后答复1733,13933)

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  Example 4
  ::例4

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  Solve the following system of equations:
  ::解决以下方程式系统:

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  $$\begin{array}{rl}5\cdot \frac{1}{x}+2\cdot \frac{1}{y}& =11\\ \frac{1}{x}+\frac{1}{y}& =4\end{array}$$

  
  ::51x+21y=111x+1y=4

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  Solution:
  ::解决方案 :

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  The strategy of elimination still applies. You can eliminate the $\begin{array}{r}\frac{1}{y}\end{array}$  term if the second equation is scaled by a factor of -2.
  ::消除战略仍然适用。如果第二个等式的乘数为-2,您可以删除一个值。如果第二个等式的乘数为-2,您可以删除一个值。

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  $$\begin{array}{rl}5\cdot \frac{1}{x}+2\cdot \frac{1}{y}& =11\\ -2\cdot \frac{1}{x}-2\cdot \frac{1}{y}& =-8\end{array}$$

  
  ::51x+21y=11-21x-21y_8

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  Add the equations together and solve for $\begin{array}{r}x\end{array}$ .
  ::x 共添加方程式并解析 。

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  $$\begin{array}{rl}3\cdot \frac{1}{x}+0\cdot \frac{1}{y}& =3\\ 3\cdot \frac{1}{x}& =3\\ \frac{1}{x}& =1\\ x& =1\end{array}$$

  
  ::31x+01y=331x=31x=1x=1

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  Substitute into the second equation and solve for $\begin{array}{r}y\end{array}$ .
  ::替代第二个方程式 并解决y。

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  $$\begin{array}{rl}\frac{1}{1}+\frac{1}{y}& =4\\ 1+\frac{1}{y}& =4\\ \frac{1}{y}& =3\\ y& =\frac{1}{3}\end{array}$$

  
  ::11+1y=41+1y=41y=3y=13

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  The point $\begin{array}{r}\left(1,\frac{1}{3}\right)\end{array}$  is the point of intersection between these two curves.
  ::点(1,13)是这两个曲线之间的交叉点。

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  Example 5
  ::例5

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  Solve the following system using elimination:
  ::使用消除方法解决以下系统:

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  $$\begin{array}{rl}11\cdot \frac{1}{x}-5\cdot \frac{1}{y}& =-38\\ 9\cdot \frac{1}{x}+2\cdot \frac{1}{y}& =-25.\end{array}$$

  
  ::111x-51y3891x+21y25。

   

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  Solution:
  ::解决方案 :

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  To eliminate $\begin{array}{r}\frac{1}{y}\end{array}$ , scale the first equation by 2 and the second equation by 5. 
  ::为了消除1y, 将第一个方程式缩放为 2, 第二个方程式缩放为 5 。

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  To eliminate $\begin{array}{r}\frac{1}{x}\end{array}$ , scale the first equation by -9 and the second equation by 11. 
  ::要消除 1x, 将第一个方程式缩放到 - 9, 第二个方程式缩放到 11 。

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  Final Answer:  $\begin{array}{r}\left(-\frac{1}{3},1\right)\end{array}$
  ::最后答复-13,1)

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  Summary
  ::摘要

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  • A system of linear equations can be solved a number of ways, including substitution, elimination, and graphing.
    ::线性方程式系统可以解决若干种方法,包括替代、消除和图形化。
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  • An independent   system is when the linear equations intersect at one point, so there is one solution to the system.
    ::独立系统是指线性方程式在某一点交叉,因此系统只有一个解决办法。
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  • A  dependent  system is when  the linear equations intersect at every point on the lines, so there are infinitely many solutions to the system.
    ::依赖系统是指线性方程式在线性线性线性线性线性线性线性每一点交叉时,因此,这个系统的解决方案无穷无尽。
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  • An  inconsistent  system is when the linear equations are parallel, so there are no solutions to the system. 
    ::不一致的系统是指线性方程式平行时,因此对系统没有解决办法。
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  • Elimination Method:
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    • Step 1 : Write both equations with two variables in standard form,  $\begin{array}{r}Ax+By=C\end{array}$ .
      ::第1步:用标准格式的两个变量(Ax+By=C)写两个方程式。
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    • Step 2 : Determine which variable you want to eliminate.
      ::第2步:确定要删除的变量。
    
    ::消除方法:第1步:以标准格式写两个变量的两个方程式,Ax+By=C。 第2步:确定要删除哪个变量。
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  • 播放段落
    • Step 3 : Scale each equation as necessary by multiplying through by constants.
      ::第3步:通过乘以常数,根据需要对每个方程式进行比例。
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    • Step 4 : Add the equations together.
      ::第4步:将方程加在一起。
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    • Step 5 :
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      • If the system is independent, solve for the first variable. Then substitute the  result in to one of the linear equations to determine the value for the second variable.  
        ::如果系统是独立的, 请解决第一个变量。 然后将结果替换为线性方程之一, 以确定第二个变量的值 。
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      • If the system is dependent, Step 4 results in  $\begin{array}{r}0=0.\end{array}$  
        ::如果系统有依赖性,第4步的结果为0=0。
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      • If the system is inconsistent, Step 4 results in  $\begin{array}{r}0=k,\end{array}$  where  $\begin{array}{r}k\end{array}$  is a nonzero number.
        ::如果系统不一致,第4步的结果为0=k,K是非零数。
      
      ::第5步:如果系统是独立的,请解决第一个变量。然后将结果替换为线性方程之一,以决定第二个变量的值。如果系统是独立的,则第4步以0=0计算。如果系统不一致,则第4步以0=k计算,K是非零数。
    
    ::第3步:通过乘以常数,根据需要对每个方程式进行比例化。第4步:将方程式加在一起。第5步:如果系统是独立的,则解决第一个变量。然后将结果替换成线性方程式之一,以确定第二个变量的值。如果系统是独立的,第4步的结果为0=0。如果系统不一致,第4步的结果为0=k,K是非零数。

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  Review
  ::回顾

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  Solve each system of equations using the elimination method.
  ::使用消除方法解决每个方程式系统。

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  1. $\begin{array}{r}x+y=-4;-x+2y=13\end{array}$
  ::1. x+y4;- x+2y=13

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  2. $\begin{array}{r}\frac{3}{2}x-\frac{1}{2}y=\frac{1}{2};-4x+2y=4\end{array}$
  ::2. 32x-12y=12;-4x+2y=4

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  3. $\begin{array}{r}6x+15y=1;2x-y=19\end{array}$
  ::3. 6x+15y=1;2x-y=19

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  4. $\begin{array}{r}x-\frac{2y}{3}=\frac{-2}{3};5x-2y=10\end{array}$
  ::4. x-2y3=10x23;5x-2y=10

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  5. $\begin{array}{r}-9x-24y=-243;\frac{1}{2}x+y=\frac{21}{2}\end{array}$
  ::5.-9x-24y243;12x+y=212

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  6. $\begin{array}{r}5x+\frac{28}{3}y=\frac{176}{3};y+x=10\end{array}$
  ::6. 5x+283y=1763;y+x=10

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  7. $\begin{array}{r}2x-3y=50;7x+8y=-10\end{array}$
  ::7. 2x-3y=50;7x+8y10

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  8. $\begin{array}{r}2x+3y=1;2y=-3x+14\end{array}$
  ::8. 2x+3y=1;2y=3x+14

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  9. $\begin{array}{r}2x+\frac{3}{5}y=3;\frac{3}{2}x-y=-5\end{array}$
  ::9. 2x+35y=3;32x-y5

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  10. $\begin{array}{r}5x=9-2y;3y=2x-3\end{array}$
  ::10. 5x=9-2y;3y=2x-3

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  11. How do you know if a system of equations has no solution?
  ::11. 你怎么知道一个方程系统是否有解决办法?

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  12. If a system of equations has no solution, what does this imply about the relationship of the curves on the graph?
  ::12. 如果一个方程式系统没有解决办法,这对图中曲线之间的关系意味着什么?

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  13. Give an example of a system of two equations with two unknowns with an infinite number of solutions. Explain how you know the system has an infinite number of solutions.
  ::13. 举一个两个方程式系统的例子,两个未知方程式,两个未知方程式有无限数量的解决办法,解释你如何知道这个系统有无限数量的解决办法。

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  14. Solve:
  ::14. 解决:

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  $$\begin{array}{rl}12\cdot \frac{1}{x}-18\cdot \frac{1}{y}& =4\\ 8\cdot \frac{1}{x}+9\cdot \frac{1}{y}& =5\end{array}$$

  
  ::121x-181y=481x+91y=5

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  15. Solve:
  ::15. 解决:

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  $$\begin{array}{rl}14\cdot \frac{1}{x}-5\cdot \frac{1}{y}& =-3\\ 7\cdot \frac{1}{x}+2\cdot \frac{1}{y}& =3\end{array}$$

  16. A 150-yard pipe is cut to provide drainage for two fields. If the length of one piece is three yards less that twice the length of the second piece, what are the lengths of the two pieces?
  ::141x-51y371x+21y=316。一个150码的管道被切开,为两个田地提供排水。如果一块长三码,比第二块长两倍,那两块的长度是多少?

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  17. Mr. Stein invested a total of $100,000 in two companies for a year. Company A's stock showed a 13% annual gain, while Company B showed a 3% loss for the year. Mr. Stein made an 8% return on his investment over the year. How much money did he invest in each company?
  ::17. Stein先生一年在两家公司总共投资100 000美元,A公司的股票年收益为13%,B公司的年收益为3%,Stein先生一年的投资回报率为8%,他在每个公司投资了多少钱?

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  18. Jack and James each buy some small fish for their new aquariums. Jack buys 10 clownfish and 7 goldfish for $28.25. James buys 5 clownfish and 6 goldfish for $17.25. How much does each type of fish cost?
  ::18. Jack和James每人为新的水族馆购买一些小鱼,Jack为28.25美元购买10条小丑鱼和7条金鱼,James为17.25美元购买5条小丑鱼和6条金鱼。

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  19. The sum of two numbers is 35. The larger number is one less than three times the smaller number. What are the two numbers?
  ::19. 两个数字的总和是35, 最大的数字是比较小的数字少1比3倍。这两个数字是多少?

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  20. Rachel offers to go to the coffee shop to buy cappuccinos and lattes for her coworkers. She buys a total of nine drinks for $35.75. If cappuccinos cost $3.75 each, and the lattes cost $4.25 each, how many of each drink did she buy?
  ::20. Rachel提议去咖啡店为同事购买卡布奇诺和拿铁,她共购买九杯饮料,价格为35.75美元,如果卡布奇诺每人价格为3.75美元,拿铁每人价格为4.25美元,她每杯要买多少?

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  Review (Answers)
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。