7.7 一致和不一致的线性系统

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  一致和不一致的线性系统

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  Consistent and Inconsistent Linear Systems 
  ::一致和不一致的线性系统

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  A system of linear equations is a set of linear equations which must be solved together. The lines in the system can be graphed together on the same coordinate graph and the solution to the system is the point at which the two lines intersect.
  ::线性方程式系统是一组线性方程式,必须一起解决。系统中的线性方程式可以在同一坐标图上一起绘制,而系统的解决办法是两条线交叉的点。

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  Or at least that’s what usually happens. But what if the lines turn out to be parallel when we graph them?
  ::或者至少这是通常会发生的事情。 但是如果我们用图表显示这些线条是平行的呢?

  

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  If the lines are parallel, they won’t ever intersect. That means that the system of equations they represent has no solution. A system with no solutions is called an inconsistent system .
  ::如果线条是平行的,它们就永远不会交叉。 这意味着它们所代表的方程式体系没有解决方案。 一个没有解决方案的体系被称为不一致的体系。

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  And what if the lines turn out to be identical?
  ::如果线条是相同的呢?

  

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  If the two lines are the same, then every point on one line is also on the other line, so every point on the line is a solution to the system. The system has an infinite number of solutions, and the two equations are really just different forms of the same equation . Such a system is called a dependent system .
  ::如果两条线相同,那么一条线上的每个点也在同一条线上,那么线上的每个点就是系统的一个解决方案。这个系统有无限数量的解决方案,而两个方程式其实只是同一方程式的不同形式。这个系统被称为依赖系统。

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  But usually, two lines cross at exactly one point and the system has exactly one solution:
  ::但通常,两条线交叉到一个点上, 而系统有一个完全的解决方案:

  

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  A system with exactly one solution is called a consistent system .
  ::一个完全只有一个解决方案的系统被称为一个一致的系统。

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  To identify a system as consistent , inconsistent , or dependent , we can graph the two lines on the same graph and see if they intersect, are parallel, or are the same line. But sometimes it is hard to tell whether two lines are parallel just by looking at a roughly sketched graph.
  ::要确定一个系统是否一致、不一致或依赖,我们可以在同一图表上绘制两条线的图表,看看两条线是交叉的,平行的,还是相同的。 但有时很难判断两条线是平行的,只是看一张大致的草图。

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  Another option is to write each line in slope-intercept form and compare the slopes and $\begin{array}{r}y-\end{array}$ intercepts of the two lines. To do this we must remember that:
  ::另一个选项是用斜坡界面形式写出每条线,比较两条线的斜坡和截截。 为此,我们必须记住:

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  • Lines with different slopes always intersect.
    ::具有不同斜坡的线条总是交叉的。
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  • Lines with the same but different $\begin{array}{r}y-\end{array}$ intercepts are parallel.
    ::具有相同但不同的 y - 界面的线条是平行的 。
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  • Lines with the same slope and the same $\begin{array}{r}y-\end{array}$ intercepts are identical.
    ::具有相同斜坡和相同y- intercuts的线是相同的。

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  Determining the Number of Solutions 
  ::确定解决方案的数量

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  1. Determine whether the following system has exactly one solution, no solutions, or an infinite number of solutions.
  ::1. 确定以下系统究竟是有一个解决办法,还是没有解决办法,还是有无限数目的解决办法。

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  $$\begin{array}{rl}2x-5y& =2\\ 4x+y& =5\end{array}$$

  
  ::2x-5y=24x+y=5

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  We must rewrite the equations so they are in slope-intercept form
  ::我们必须重写方程, 使方程以斜坡界面形式出现

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  $\begin{array}{r}2x-5y=2\Rightarrow -5y=-2x+2\Rightarrow y=\frac{2}{5}x-\frac{2}{5}\end{array}$
  ::2x- 5y=25y2x+25y=25x- 25

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  $\begin{array}{r}4x+y=5\Rightarrow y=-4x+5\end{array}$
  ::4x+y=5y4x+5

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  The slopes of the two equations are different; therefore the lines must cross at a single point and the system has exactly one solution. This is a consistent system.
  ::这两个方程的斜坡不同;因此,线条必须在一个点交叉,而系统完全只有一个解决办法。这是一个一致的系统。

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  2. Determine whether the following system has exactly one solution, no solutions, or an infinite number of solutions.
  ::2. 确定以下系统究竟是有一个解决办法,还是没有解决办法,还是有无限数目的解决办法。

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  $$\begin{array}{rl}3x& =5-4y\\ 6x+8y& =7\end{array}$$

  
  ::3x=5-4y6x+8y=7

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  We must rewrite the equations so they are in slope-intercept form
  ::我们必须重写方程, 使方程以斜坡界面形式出现

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  $\begin{array}{r}3x=5-4y\Rightarrow 4y=-3x+5\Rightarrow y=-\frac{3}{4}x+\frac{5}{4}\end{array}$
  ::3x=5-4y4y3x+54x+54

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  $\begin{array}{r}6x+8y=7\Rightarrow 8y=-6x+7\Rightarrow y=-\frac{3}{4}x+\frac{7}{8}\end{array}$
  ::6x+8y=78y6x+74x78

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  The slopes of the two equations are the same but the $\begin{array}{r}y-\end{array}$ intercepts are different; therefore the lines are parallel and the system has no solutions. This is an inconsistent system.
  ::两个方程式的斜坡是相同的,但y - 截面是不同的;因此,线条是平行的,系统没有解决办法。这是一个不一致的系统。

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  3. Determine whether the following system has exactly one solution, no solutions, or an infinite number of solutions.
  ::3. 确定以下系统究竟是有一个解决办法,还是没有解决办法,还是有无限数目的解决办法。

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

  
  ::x+y=33x+3y=9

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  We must rewrite the equations so they are in slope-intercept form
  ::我们必须重写方程, 使方程以斜坡界面形式出现

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

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  $\begin{array}{r}3x+3y=9\Rightarrow 3y=-3x+9\Rightarrow y=-x+3\end{array}$
  ::3x+3y=9\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\+3\\\\\\\\\\\\\\\\\\\\ x+3\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\3\\\\\\\\\\\\\\3\\\\\\\\\\\\\\\\\\

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  The lines are identical; therefore the system has an infinite number of solutions. It is a dependent system.
  ::线条相同; 因此, 系统有无限数的解决方案。 它是一个依赖系统 。

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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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  Determine whether the following system of linear equations has zero, one, or infinitely many solutions:
  ::确定下列线性方程式系统是否为零、一或无限多的解决方案:

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  $$\begin{array}{r}\{\begin{array}{l}2y+6x=20\\ y=-3x+7\end{array}\end{array}$$

  
  ::{2y+6x=20y3x+7

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  What kind of system is this?
  ::这是什么系统?

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  It is easier to compare equations when they are in the same form. We will rewrite the first equation in slope-intercept form.
  ::比较相同形式的方程式比较容易。 我们将重写第一个方程式, 以斜坡界面形式 。

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  $\begin{array}{r}2y+6x=20\Rightarrow y+3x=10\Rightarrow y=-3x+10\end{array}$
  ::2 y+6x=20 y+3x=10 y=3x+10

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  Since the two equations have the same slope, but different $\begin{array}{r}y\end{array}$ -intercepts, they are different but parallel lines. Parallel lines never intersect, so they have no solutions.
  ::由于两个方程式具有相同的斜坡,但不同的 Y 界面,它们是不同的但平行的线条。平行的线条从不交叉,因此它们没有解决办法。

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  Since the lines are parallel, it is an inconsistent system.
  ::由于线条是平行的,它是一个不一致的制度。

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  Review 
  ::回顾

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  Express each equation in slope-intercept form. Without graphing, state whether the system of equations is consistent, inconsistent or dependent.
  ::以斜坡截取形式表达每个方程式。 不绘制图形, 请说明方程式系统是否一致、 不一致或依赖 。

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  1. $$\begin{array}{r}3x-4y=13\\ y=-3x-7\end{array}$$
     
     ::3x-4y=13y3x-7
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  2. $$\begin{array}{r}\frac{3}{5}x+y=3\\ 1.2x+2y=6\end{array}$$
     
     ::35x+y=31.2x+2y=6
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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. $$\begin{array}{r}3x-3y=3\\ x-y=1\end{array}$$
     
     ::3x-3y=3x-y=1
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  5. $$\begin{array}{r}0.5x-y=30\\ 0.5x-y=-30\end{array}$$
     
     ::0.5x-y=300.5x-y30
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  6. $$\begin{array}{r}4x-2y=-2\\ 3x+2y=-12\end{array}$$
     
     ::4 - 2y 23x+2y 12
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  7. $$\begin{array}{r}3x+y=4\\ y=5-3x\end{array}$$
     
     ::3x+y=4y=5-3x
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  8. $$\begin{array}{r}x-2y=7\\ 4y-2x=14\end{array}$$
     
     ::x-2y=74y-2x=14
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  9. $$\begin{array}{rl}& -2y+4x=8\\ & y-2x=-4\end{array}$$
     
     ::-2y+4x=8y-2x=4
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  10. $$\begin{array}{rl}& x-\frac{y}{2}=\frac{3}{2}\\ & 3x+y=6\end{array}$$
      
      ::x- y2=323x+y=6
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  11. $$\begin{array}{rl}& 0.05x+0.25y=6\\ & x+y=24\end{array}$$
      
      ::0.05x+0.25y=6x+y=24
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  12. $$\begin{array}{rl}& x+\frac{2y}{3}=6\\ & 3x+2y=2\end{array}$$
      
      ::x+2y3=63x+2y=2

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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 。