7.6 解决线性系统的比较方法

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  解决线性系统的比较方法

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  Comparing Methods for Solving Linear Systems 
  ::解决线性系统的比较方法

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  There are a number of methods for solving systems of equations, and each has its own strengths. F or simplicity, we’ll look at them in table form. This should help you decide which method would be best for a given situation.
  ::解答方程式系统的方法有很多种,每个方程式都有其自身的优点。 为了简单起见,我们将以表格的形式审视这些方程式。 这应该有助于您决定哪种方法最适合特定情况。

  Method:	Best used when you...	Advantages:	Comment:
  Graphing	...don’t need an accurate answer.	Often easier to see number and quality of intersections on a graph. With a graphing calculator, it can be the fastest method since you don’t have to do any computation.	Can lead to imprecise answers with non- integer solutions.
  Substitution	...have an explicit equation for one variable (e.g. $\begin{array}{r}y=14x+2\end{array}$ )	Works on all systems. Reduces the system to one variable, making it easier to solve.	You are not often given explicit functions in systems problems, so you may have to do extra work to get one of the equations into that form.
  Elimination by Addition or Subtraction	...have matching coefficients for one variable in both equations.	Easy to combine equations to eliminate one variable. Quick to solve.	It is not very likely that a given system will have matching coefficients.
  Elimination by Multiplication and then Addition and Subtraction	...do not have any variables defined explicitly or any matching coefficients.	Works on all systems. Makes it possible to combine equations to eliminate one variable.	Often more algebraic manipulation is needed to prepare the equations.

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  The table above is only a guide. You might prefer to use the graphical method for every system in order to better understand what is happening, or you might prefer to use the multiplication method even when a substitution would work just as well.
  ::上表只是一个指南。 您可能更喜欢使用每个系统的图形方法, 以便更好地了解正在发生的情况, 或者您可能更喜欢使用乘法, 即使替代方法同样有效 。

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  Solve by Graphing  
  ::通过图形绘制解决

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  Two angles are complementary when the sum of their angles is $\begin{array}{r}{90}^{\circ }\end{array}$ . Angles $\begin{array}{r}A\end{array}$ and $\begin{array}{r}B\end{array}$ are complementary angles, and twice the measure of angle $\begin{array}{r}A\end{array}$ is $\begin{array}{r}{9}^{\circ }\end{array}$ more than three times the measure of angle $\begin{array}{r}B\end{array}$ . Find the measure of each angle.
  ::当角度之和为 90 时,两个角度是互补的。 角度 A 和 B 是互补的角, 角度 A 的测量值是 9 的两倍, 是角度 B 的测量值的三倍以上。 查找每个角度的测量值 。

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  First we write out our 2 equations. We will use $\begin{array}{r}x\end{array}$ to be the measure of angle $\begin{array}{r}A\end{array}$ and $\begin{array}{r}y\end{array}$ to be the measure of angle $\begin{array}{r}B\end{array}$ . We get the following system:
  ::首先我们写出我们的两个方程式。 我们将使用 x 来测量角度 A 和 y 。 我们得到以下系统 :

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  $$\begin{array}{rl}x+y& =90\\ 2x& =3y+9\end{array}$$

  
  ::x+y=902x=3y+9

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  First, we’ll solve this system with the graphical method. For this, we need to convert the two equations to $\begin{array}{r}y=mx+b\end{array}$ form:
  ::首先,我们将用图形方法解决这个系统。 为此, 我们需要将两个方程式转换为 y=mx+b 窗体 :

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  $$\begin{array}{rl}& x+y=90\Rightarrow y=-x+90\\ & 2x=3y+9\Rightarrow y=\frac{2}{3}x-3\end{array}$$

  
  ::x+y=90 yx+902x=3y+9y=23x-3

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  The first line has a of -1 and a $\begin{array}{r}y-\end{array}$ intercept of 90, and the second line has a slope of $\begin{array}{r}\frac{2}{3}\end{array}$ and a $\begin{array}{r}y-\end{array}$ intercept of -3. The graph looks like this:
  ::第一行有 -1 和 Y - interview 90, 第二行有 23 的斜度和 Y - interview 3 的 y - interview。

  

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  In the graph, it appears that the lines cross at around $\begin{array}{r}x=55,y=35\end{array}$ , but it is difficult to tell exactly! Graphing by hand is not the best method in this case!
  ::在图表中,线条在x=55,y=35左右交叉,但很难确切辨别!用手绘图不是本案的最佳方法!

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  Solving by Substitution 
  ::以替代方式解决

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  In this example, we’ll try solving by substitution. Let’s look again at the system:
  ::以这个例子来说,我们将尝试用替代来解决。 让我们再看看这个系统:

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  $$\begin{array}{rl}x+y& =90\\ 2x& =3y+9\end{array}$$

  
  ::x+y=902x=3y+9

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  We’ve already seen that we can start by solving either equation for $\begin{array}{r}y\end{array}$ , so let’s start with the first one:
  ::我们已看到我们可以从解决y这两个方程式开始, 所以让我们先从第一个方程式开始:

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  $$\begin{array}{r}y=90-x\end{array}$$

  
  ::y=90-x

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  Substitute into the second equation:
  ::替代第二个方程:

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  $$\begin{array}{rlrl}& 2x=3(90-x)+9& & \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{e}3\\ & 2x=270-3x+9& & \mathrm{a}\mathrm{d}\mathrm{d}3\mathrm{x}\mathrm{t}\mathrm{o}\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{s}\\ & 5x=270+9=279& & \mathrm{d}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{b}\mathrm{y}5\\ & x={55.8}^{\circ }\end{array}$$

  
  ::2x=3(90-x)+9 将 32x=270-3x+9add 3x 分配到两侧5x=270+9=279divide 5x=55.8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Substitute back into our expression for $\begin{array}{r}y\end{array}$ :
  ::取代我们表达的y:

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  $$\begin{array}{r}y=90-55.8={34.2}^{\circ }\end{array}$$

  
  ::y=90-55.8=34.2

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  Angle $\begin{array}{r}A\end{array}$ measures $\begin{array}{r}{55.8}^{\circ }\end{array}$ ; angle $\begin{array}{r}B\end{array}$ measures $\begin{array}{r}{34.2}^{\circ }\end{array}$ .
  ::角度AgleA量度55.8;角度B量度34.2。

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  Solving by Elimination 
  ::通过消除解决

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  Finally, in this example, we’ll try solving by elimination (with multiplication):
  ::最后,在这个例子中,我们将尝试通过消除(乘以乘法)来解决:

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  Rearrange equation one to standard form :
  ::将方程 1 重新排列为标准格式 :

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  $$\begin{array}{rl}& x+y=90\Rightarrow 2x+2y=180\end{array}$$

  
  ::x+y=902x+2y=180

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  Multiply equation two by 2:
  ::乘以二乘以二乘以二乘以二

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

  
  ::2x=3y+92x-3y=9

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  Subtract:
  ::减:

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  $$\begin{array}{rl}& 2x+2y=180\\ & -(2x-3y)=-9\\ & \underset{\_}{}\\ & 5y=171\\ \\ & \text{Divide by 5 to obtain}y={34.2}^{\circ }\end{array}$$

  
  ::2x+2y=180 -(2x-3y)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\DDDDD555555DDDD5555\5DD5\5\5\5DDD5\\\5\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Substitute this value into the very first equation:
  ::将这个值替换为第一个方程:

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  $$\begin{array}{rlrl}x+34.2& =90& & \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}34.2\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{s}\\ x& ={55.8}^{\circ }\end{array}$$

  
  ::x+34.2=90 从两侧的34.2次数=55.8

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  Angle $\begin{array}{r}A\end{array}$ measures $\begin{array}{r}{55.8}^{\circ }\end{array}$ ; angle $\begin{array}{r}B\end{array}$ measures $\begin{array}{r}{34.2}^{\circ }\end{array}$ .
  ::角 A 度量55.8;角 B 度量34.2。

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  Even though this system looked ideal for substitution, the method of multiplication worked well too. Once the equations were rearranged properly, the solution was quick to find. You’ll need to decide yourself which method to use in each case you see from now on. Try to master all the techniques, and recognize which one will be most efficient for each system you are asked to solve.
  ::即使这个系统看起来是理想的替代方法,但乘法也很有效。 一旦方程式重新排列得当,解决方案就会很快找到。 从现在开始,你需要自己决定在每种情况下使用哪种方法。 尝试掌握所有的技术,并承认哪种方法对每个系统最有效,你被要求解决。

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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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  Solve the system $\begin{array}{r}\{\begin{array}{l}5s+2t=6\\ 9s+2t=22\end{array}\end{array}$
  ::解决系统 {5s+2t=69s+2t=22

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  Since these equations are both written in standard form, and both have the term $\begin{array}{r}2t\end{array}$ in them, we will use elimination by subtracting. This will cause the $\begin{array}{r}t\end{array}$ terms to cancel out and we will be left with one variable, $\begin{array}{r}s\end{array}$ , which we can then isolate.
  ::由于这两个方程式都是以标准形式写成的,而且两者都含有第2个术语,我们将用减去来消除这些方程式。 这将导致 t 条件取消, 并留下一个变量, s, 然后我们就可以分离出来。

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  $$\begin{array}{rl}& 5s+2t=6\\ & \underset{\_}{-(9s+2t=22)}\\ & -4s+0t=-16\\ & -4s=-16\\ & s=4\end{array}$$

  
  ::5s+2t=6-(9s+2t=22)-(9s+2t=22)-(4s+0t16)-(4s=4)

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  $$\begin{array}{rl}5(4)+2t& =6\\ 20+2t& =6\\ 2t& =-14\\ t& =-7\end{array}$$

  
  ::5(4)+2t=620+2t=62t14t7

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  The solution is $\begin{array}{r}(4,-7)\end{array}$ .
  ::解决办法是(4,-7)。

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  Review 
  ::回顾

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  Solve the following systems using any method.
  ::使用任何方法解决下列系统。

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  1. $$\begin{array}{r}x=3y\\ x-2y=-3\end{array}$$
     
     ::x=3yx-2y3
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  2. $$\begin{array}{r}y=3x+2\\ y=-2x+7\end{array}$$
     
     ::y=3x+2y=2x+7 y=3x+2y=2x+7
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  3. $$\begin{array}{r}5x-5y=5\\ 5x+5y=35\end{array}$$
     
     ::5x-5y=55x+5y=35
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  4. $$\begin{array}{r}y=-3x-3\\ 3x-2y+12=0\end{array}$$
     
     ::y3x- 33x- 2y+12=0
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  5. $$\begin{array}{r}3x-4y=3\\ 4y+5x=10\end{array}$$
     
     ::3x-4y=34y+5x=10
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  6. $$\begin{array}{r}9x-2y=-4\\ 2x-6y=1\end{array}$$
     
     ::9 - 2y% 42x-6y=1
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  7. Supplementary angles are two angles whose sum is $\begin{array}{r}{180}^{\circ }\end{array}$ . Angles $\begin{array}{r}A\end{array}$ and $\begin{array}{r}B\end{array}$ are supplementary angles. The measure of Angle $\begin{array}{r}A\end{array}$ is $\begin{array}{r}{18}^{\circ }\end{array}$ less than twice the measure of Angle $\begin{array}{r}B\end{array}$ . Find the measure of each angle.
     ::补充角度是两个角度,其总和为 180 。 角度 A 和 B 是 补充角度。 角度 A 的度量为 18 , 低于 角度 B 的度量的两倍。 查找每个角度的度量 。
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  8. A farmer has fertilizer in 5% and 15% solutions. How much of each type should he mix to obtain 100 liters of fertilizer in a 12% solution?
     ::农民有百分之五和百分之十五的化肥解决方案。 每类化肥中,他应该混合多少来获得百分之十二的化肥?
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  9. 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?
     ::150码的管道被切断,为两个田地提供排水。如果一块长3码,比第二块长两倍,那么两块的长度是多少?
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  10. 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?
      ::Stein先生一年在两家公司总共投资10万美元。 A公司的股票年收益为13 % , B公司年收益为3 % 。 Stein先生全年投资回报率为8 % 。 他对每家公司投资了多少钱?

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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 。