7.8 确定线性系统类型

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• 选择节确定线性系统类型

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  确定线性系统类型

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  Determining the Type of Linear System 
  ::确定线性系统类型

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  A third option for identifying systems as consistent , inconsistent or dependent is to just solve the system and use the result as a guide.
  ::确定系统是否一致、不一致或依赖性的第三种选择是,只解决系统问题,并将结果作为指南使用。

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  Consistent Systems 
  ::统一系统

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  Solve the following system of equations . Identify the system as consistent, inconsistent or dependent.
  ::解决以下方程式系统。 将系统识别为一致、 不一致或依赖性 。

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  $$\begin{array}{rl}10x-3y& =3\\ 2x+y& =9\end{array}$$

  
  ::10x-3y=32x+y=9

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  Let’s solve this system using the substitution method.
  ::让我们使用替代方法来解决这个系统。

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  Solve the second equation for $\begin{array}{r}y\end{array}$ :
  ::解决y的第二个方程式:

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  $$\begin{array}{r}2x+y=9\Rightarrow y=-2x+9\end{array}$$

  
  ::2x+y=9y%2x+9

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  Substitute that expression for $\begin{array}{r}y\end{array}$ in the first equation:
  ::替换第一个方程式中的 y 表达式 :

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  $$\begin{array}{rl}10x-3y& =3\\ 10x-3(-2x+9)& =3\\ 10x+6x-27& =3\\ 16x& =30\\ x& =\frac{15}{8}\end{array}$$

  
  ::10x-3y=310x-3(-2x+9)=310x-6x-27=316x=30x=158

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  Substitute the value of $\begin{array}{r}x\end{array}$ back into the second equation and solve for $\begin{array}{r}y\end{array}$ :
  ::将 x 的值替换回第二个方程式, 并解决 y:

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  $$\begin{array}{r}2x+y=9\Rightarrow y=-2x+9\Rightarrow y=-2\cdot \frac{15}{8}+9\Rightarrow y=\frac{21}{4}\end{array}$$

  
  ::2x+y=9 *2x+9 *2________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  The solution to the system is $\begin{array}{r}\left(\frac{15}{8},\frac{21}{4}\right)\end{array}$ . The system is consistent since it has only one solution.
  ::系统的解决办法是(158 214),系统是一致的,因为它只有一个解决办法。

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  Inconsistent Systems 
  ::不一致的系统

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  Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.
  ::解决以下方程式系统。 将系统识别为一致、 不一致或依赖性 。

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  $$\begin{array}{rl}3x-2y& =4\\ 9x-6y& =1\end{array}$$

  
  ::3x-2y=49x-6y=1

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  Let’s solve this system by the method of multiplication .
  ::让我们用乘法解决这个系统。

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  Multiply the first equation by 3:
  ::乘以第一个方程乘以 3 :

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  $$\begin{array}{r}3(3x-2y=4)9x-6y=12\\ \Rightarrow \\ 9x-6y=19x-6y=1\end{array}$$

  
  ::3(3x--2y=4)9x-6y=12*9x-6y=1 9x-6y=1

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  Add the two equations:
  ::添加两个方程式:

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  $$\begin{array}{rl}& 9x-6y=4\\ & \underset{\_}{9x-6y=1}\\ & 0=13\text{This statement is not true.}\end{array}$$

  
  ::9 - 6y=49x- 6y=1_ 0=13 此语句不正确 。

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  If our solution to a system turns out to be a statement that is not true, then the system doesn’t really have a solution; it is inconsistent.
  ::如果我们对一个系统的解决方案被证明是一个不真实的声明,那么这个系统其实没有真正的解决方案;它就是不一致的。

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  Dependent Systems 
  ::依赖系统

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  Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.
  ::解决以下方程式系统。 将系统识别为一致、 不一致或依赖性 。

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  $$\begin{array}{rl}4x+y& =3\\ 12x+3y& =9\end{array}$$

  
  ::4x+y=312x+3y=9

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  Let’s solve this system by substitution.
  ::让我们用替代来解决这个制度。

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  Solve the first equation for $\begin{array}{r}y\end{array}$ :
  ::解决y的第一个方程式:

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  $$\begin{array}{r}4x+y=3\Rightarrow y=-4x+3\end{array}$$

  
  ::4x+y=3y4x+3

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  Substitute this expression for $\begin{array}{r}y\end{array}$ in the second equation:
  ::在第二个方程式中替换 Y 的这个表达式 :

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  $$\begin{array}{rl}12x+3y& =9\\ 12x+3(-4x+3)& =9\\ 12x-12x+9& =9\\ 9& =9\end{array}$$

  
  ::12x+3y=912x3(- 4x+3)=912x- 12x+9=99=9

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  This statement is always true.
  ::这一说法始终是真实的。

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  If our solution to a system turns out to be a statement that is always true, then the system is dependent.
  ::如果我们对一个系统的解决方案 被证明是一个声明 永远是真实的, 那么这个系统是依附的。

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  A second glance at the system in this example reveals that the second equation is three times the first equation, so the two lines are identical. The system has an infinite number of solutions because they are really the same equation and trace out the same line.
  ::对这一示例中的系统进行第二眼观察后发现,第二个方程是第一个方程的三倍,所以两行是相同的。这个系统有无限数量的解决方案,因为它们是完全相同的方程,并追踪同一行。

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  Let’s clarify this statement. An infinite number of solutions does not mean that any ordered pair $\begin{array}{r}(x,y)\end{array}$ satisfies the system of equations. Only ordered pairs that solve the equation in the system (either one of the equations) are also solutions to the system. There are infinitely many of these solutions to the system because there are infinitely many points on any one line.
  ::让我们来澄清这个语句。 无限数量的解决方案并不意味着任何定购对(x,y)都符合方程式体系。 只有解决系统中方程式的定购对(两个方程式之一)才是系统的解决办法。 系统有许多这样的解决方案,因为任何一个行都有无限多的点。

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  For example, (1, -1) is a solution to the system in this example, and so is (-1, 7). Each of them fits both the equations because both equations are really the same equation. But (3, 5) doesn’t fit either equation and is not a solution to the system.
  ::比如,在这个例子中,(1,1)和(1,7)都是系统的一种解决方案,而(1,7)和(1,7)都符合两个方程,因为两个方程其实是相同的方程。 但(3,5)两个方程都不符合两个方程,也不是系统的解决办法。

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  In fact, for every $\begin{array}{r}x-\end{array}$ value there is just one $\begin{array}{r}y-\end{array}$ value that fits both equations, and for every $\begin{array}{r}y-\end{array}$ value there is exactly one $\begin{array}{r}x-\end{array}$ value—just as there is for a single line.
  ::事实上,对于每一个x-价值,只有一个y-价值适合两个方程,对于每一个y-价值,就有一个x-价值——就像单行一样。

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  Example
  ::示例示例示例示例

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  Example 1
  ::例1

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  Identify the system as consistent, inconsistent, or consistent-dependent.
  ::将该系统确定为一致、不一致或依赖一致的系统。

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  $$\begin{array}{rl}3x-2y& =4\\ 9x-6y& =1\end{array}$$

  
  ::3x-2y=49x-6y=1

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  Solution: Because both equations are in standard form , elimination is the best method to solve this system.
  ::解决方案:因为两个方程式都是标准形式,消除是解决这个系统的最佳方法。

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  Multiply the first equation by 3.
  ::乘以第一个方程乘以3。

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  $$\begin{array}{rlr}3(3x-2y=4)& & 9x-6y=12\\ & \Rightarrow & \\ 9x-6y=1& & 9x-6y=1\end{array}$$

  
  ::3(3x--2y=4)9x-6y=12*9x-6y=19x-6y=1

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  Subtract the two equations.
  ::减去两个方程。

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  $$\begin{array}{rl}& 9x-6y=12\\ & \underset{\_}{9x-6y=1}\\ & 0=11\text{This Statement is not true.}\end{array}$$

  
  ::9- 6y=129x- 6y=1_ 0=11 此语句不正确 。

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  This is an untrue statement; therefore , you can conclude:
  ::这是不真实的陈述;因此,你可以得出以下结论:

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  1. These lines are parallel.
     ::这些线是平行的。
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  2. The system has no solution.
     ::该系统没有解决办法。
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  3. The system is inconsistent.
     ::该系统前后不一致。

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  Review 
  ::回顾

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  Find the solution of each system of equations using the method of your choice. State if the system is inconsistent or dependent.
  ::使用您选择的方法,找到每种方程的解决方案。如果系统不一致或依赖,请说明。

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  1. $$\begin{array}{r}3x+2y=4\\ -2x+2y=24\end{array}$$
     
     ::3x+2y=4-2x+2y=24
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  2. $$\begin{array}{r}5x-2y=3\\ 2x-3y=10\end{array}$$
     
     ::5x-2y=32x-3y=10
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  3. $$\begin{array}{r}3x-4y=13\\ y=-3x-7\end{array}$$
     
     ::3x-4y=13y3x-7
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  4. $$\begin{array}{r}5x-4y=1\\ -10x+8y=-30\end{array}$$
     
     ::5x-4y=1-10x+8y30
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  5. $$\begin{array}{r}4x+5y=0\\ 3x=6y+4.5\end{array}$$
     
     ::4x+5y=03x=6y+4.5
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  6. $$\begin{array}{r}-2y+4x=8\\ y-2x=-4\end{array}$$
     
     ::-2y+4x=8y-2x=4
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  7. $$\begin{array}{r}x-\frac{1}{2}y=\frac{3}{2}\\ 3x+y=6\end{array}$$
     
     ::x- 12y=323x+y=6
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  8. $$\begin{array}{r}0.05x+0.25y=6\\ x+y=24\end{array}$$
     
     ::0.05x+0.25y=6x+y=24
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  9. $$\begin{array}{r}x+\frac{2}{3}y=6\\ 3x+2y=2\end{array}$$
     
     ::x+23y=63x+2y=2
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  10. $$\begin{array}{rl}& 3x-4y=13\\ & y=-3x-7\end{array}$$
      
      ::3x-4y=13y3x-7
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  11. $$\begin{array}{rl}& 4x+y=3\\ & 12x+3y=9\end{array}$$
      
      ::4x+y=312x+3y=9
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  12. $$\begin{array}{rl}& 10x-3y=3\\ & 2x+y=9\end{array}$$
      
      ::10x-3y=32x+y=9

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  Review (Answers)
  ::回顾(答复)

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  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 。