8.1 两种等同和两种未知的系统

Section outline

• Select section General

  Collapse Expand

  General

  Collapse all Expand all

  • Select activity 新闻通告

    

    新闻通告 Forum
• Select section 两个等量和两个未知星系

  Collapse Expand

  两个等量和两个未知星系

  

  播放段落

  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 .  As a consumer, it would be useful to know when the two plans cost the same and when one plan is cheaper. 
  ::两个手机计划的费用可以写成一个方程式系统,以所用分钟数和每月基本费率为基础,作为消费者,最好能知道这两个计划的费用是多少,一个计划的费用是多少。

  播放段落

  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。

  播放段落

  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。

  播放段落

  Plan B has a lower starting cost, but since it costs more per minute, it may not be the right plan for someone who likes to spend a lot of time on the phone.  When do the two plans cost the same amount?
  ::B计划起点成本较低,但是由于每分钟成本更高,对于喜欢在电话上花很多时间的人来说,它可能不是正确的计划。 这两个计划何时成本相同?

  播放段落

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

  播放段落

  There are many ways to solve a system that you have learned in the past including substitution and graphical intersection .  Here you will focus on solving using elimination because the knowledge and skills used will transfer directly into using matrices. 
  ::有很多方法可以解决你过去学到的系统, 包括替代和图形交叉点。 在这里, 您将集中解决消除问题, 因为使用的知识和技能将直接转移到使用矩阵中 。

  播放段落

  When solving a system, the first thing to do is to 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.  Two equations and two variables can be solved, but one equation with two variables cannot. 
  ::当解决一个系统时,首先要做的是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。两个方程和两个变量可以解答,但一个方程加上两个变量则无法解答。

  播放段落

  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:

  播放段落
  • 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。
  播放段落
  • Step 2 : Determine which variable you want to eliminate.
    ::第2步:确定要删除的变量。
  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。
  播放段落
  • Step 4 : Add the equations together. This should reduce both the number of equations and the number of variables leaving one equation and one variable.
    ::第4步:将方程式加在一起。这将减少方程式的数量和保留一个方程式和一个变量的变量的数量。
  播放段落
  • Step 5 : Solve and substitute to determine the value of the second variable.
    ::第5步:解决并替换确定第二个变量值的第二个变量值。

  播放段落

  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 that there is an  $\begin{array}{r}x\end{array}$ column and a  $\begin{array}{r}y\end{array}$ column on the left hand side and a constant column on the right hand side when you rewrite the equations as shown.  Also notice that if you add the system as written no variable will be eliminated. 
  ::这是由两个方程式和两个变量组成的系统, 标准格式为 5x+12y= 72and3x- 2y=18。 请注意, 当您重写所显示的方程式时, 左手侧有一个 x 列和一个 y 列, 右手侧有一个 常数列 。 另请注意, 如果您将系统添加成文字, 将不会删除变量 。

  播放段落

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

  播放段落

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

  播放段落

  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.
  ::战略性地选择通过将第二个方程缩放6来消除y,使y的系数与12和12相匹配。

  播放段落

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

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

  播放段落

  Add the two equations:
  ::添加两个方程式:

  播放段落

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

  
  ::23x=180x=18023

  播放段落

  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 will be easier to repeat the elimination process in order to solve for $\begin{array}{r}x\end{array}$ .  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式中的任何一个方程式,结果可以为y解答;然而,由于该值是一个分数,因此更容易重复清除过程,以便解答 x。 这次你将采用前两个方程式,通过将 x 的系数定为 15 和 - 15 来消除 x。 将第一个方程式缩放为 3 倍,将第二个方程式缩放为 - 5 倍。

  播放段落

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

  播放段落

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

  播放段落

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

  播放段落

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

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

  播放段落

  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。

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Earlier, you were asked about two phone plans.
  ::之前有人问过你两个电话计划

  播放段落

  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。

  播放段落

  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。

  播放段落

  If you want to find out when the two plans cost the same, you can 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.
  ::如果您想要知道这两个计划的成本何时相同, 您可以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 。

  播放段落

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

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

  播放段落

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

  播放段落

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

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

  播放段落

  Then you scale the second equation by -1 and 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, 并添加方程, 并为 y 解决问题 。

  播放段落

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

  
  ::-8 -350y=43.75

  播放段落

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

  播放段落

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

  
  ::-4x=150x=37.5

  播放段落

  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。

  播放段落

  Example 2
  ::例2

  播放段落

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

  播放段落

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

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

  播放段落

  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 。

  播放段落

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

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

  播放段落

  The sum is:
  ::总和是:

  播放段落

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

  
  ::3x=5x=53

  播放段落

  You can substitute  $\begin{array}{r}x\end{array}$ into the first equation to solve for $\begin{array}{r}y\end{array}$ .
  ::您可以将 x 替换为第一个方程式, 以解决 y 。

  播放段落

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

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

  播放段落

  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。

  播放段落

  Example 3
  ::例3

  播放段落

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

  播放段落

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

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

  播放段落

  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.
  ::开始将第一个方程式缩放7, 并注意当对等方程式进行总和时, Y 系数将立即取消 。

  播放段落

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

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

  播放段落

  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, 解答 x= 173333。 实践不是取代 x, 而是通过将第一个方程缩放为 2 和第二个方程缩放为 5 来消除 x 。

  播放段落

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

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

  播放段落

  Add, solve for $\begin{array}{r}y\end{array}$ . 
  ::添加,解决y。

  播放段落

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

  播放段落

  Example 4
  ::例4

  播放段落

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

  播放段落

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

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

  播放段落

  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。

  播放段落

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

  
  ::51x+21y=11-2(1x)-2(1y)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

  播放段落

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

  播放段落

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

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

  播放段落

  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。

  播放段落

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

  
  ::1(1)+1y=41+1y=41y=3y=13

  播放段落

  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。

  播放段落

  Example 5
  ::例5

  播放段落

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

  播放段落

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

  
  ::111x-51y3891x+21y25

  播放段落

  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 。

  播放段落

  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 。

  播放段落

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

  Summary

  播放段落 To solve a system of equations using the elimination method 播放段落 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。 播放段落 Step 2: Determine which variable you want to eliminate. ::第2步:确定要删除的变量。 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。 播放段落 Step 4: Add the equations together,  leaving one equation and one variable. ::第4步:将方程加在一起,留下一个方程和一个变量。 播放段落 Step 5: Solve and substitute to determine the value of the second variable. ::第5步:解决并替换确定第二个变量值的第二个变量值。 ::要用消除方法解决方程式系统 步骤1:用标准格式写两个变量的两种方程式,Ax+By=C. 步骤2:确定要删除的变量。步骤3:根据需要通过乘以常数对每个方程式进行缩放。步骤4:将方程式加在一起,留下一个方程式和一个变量。步骤5:解决和替换确定第二个变量的价值。

  播放段落

  Review
  ::回顾

  播放段落

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

  播放段落

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

  播放段落

  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

  播放段落

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

  播放段落

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

  播放段落

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

  播放段落

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

  播放段落

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

  播放段落

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

  播放段落

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

  播放段落

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

  播放段落

  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?

  播放段落

  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?

  播放段落

  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。

  播放段落

  14. Solve:
  ::14. 解决:

  播放段落

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

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

  播放段落

  15. Solve:
  ::15. 解决:

  播放段落

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

  
  ::141x-51y371x+21y=3

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。