Course: 3.5 采用替代办法的溶解系统无或无限多的溶解方法

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使用替代方法的无或无限多种解决办法的解决系统
Paran's cell phone company charges a flat rate of $25 per month plus $0.25 per text. Marcel's cell phone company charges a flat rate of $100 and $1 per text. Marcel's bill for the month is four times Paran's. If they sent the same number of texts, how many did they each send?
::Marcel的手机公司每月收费为100美元和1美元。Marcel的每月账单是Paren的4倍。如果他们发送的文本数量相同,他们每人发送了多少份?

Systems with No or Infinitely Many Solutions Using Substitution
::使用替代方法的系统无或无限多的系统

When a system has no solution or an infinite number of solutions and we attempt to find a single, unique solution using an algebraic method, such as substitution, the variables will cancel out and we will have an equation consisting of only constants. If the equation is untrue then the system has no solution. If the equation is always true then there are infinitely many solutions.
::当一个系统没有解决方案或无限数的解决方案,而我们试图用代数法(如替代)找到一个单一、独特的解决方案时,变量就会取消,我们就会有一个由常数组成的方程式。如果方程式不真实,那么系统就没有解决方案。如果方程式总是对的,那么就有很多解决方案。

Let's solve the following .
::让我们解决以下问题。

$\begin{aligned} 3 x - 2 y & = 7 \\ y & = \frac{3}{2} x + 5 \end{aligned}$
$\begin{align*}3x-2y &= 7\\ y &= \frac{3}{2}x+5\end{align*}$
::3x-2yy=7y=32x+5

Since the second equation is already solved for $\begin{align*}y\end{align*}$ , we can use this in the first equation to solve for $\begin{align*}x\end{align*}$ :
::由于y的第二个方程式已经解决了, 我们可以在第一个方程式中用它来解决 x:

$\begin{aligned} 3 x - 2 (\frac{3}{2} x + 5) & = 7 \\ 3 x - 3 x - 10 & = 7 \\ - 10 & \neq 7 \end{aligned}$
$\begin{align*}3x-2 \left(\frac{3}{2}x+5\right) &= 7\\ 3x-3x-10 &= 7\\ \text{-}10 & \ne 7\end{align*}$
::3x-2(32x+5)=73x-3x-10=7-107

Since the substitution above resulted in the elimination of the variable , $\begin{align*}x\end{align*}$ , and an untrue equation involving only constants, the system has no solution. The lines are parallel and the system is inconsistent .
::由于上述替代导致变数、x和不真实的方程式被删除,只涉及常数,因此系统没有解决办法,线条是平行的,系统是不一致的。

Let's solve the following systems using substitution.
::让我们用替代解决以下系统

$\begin{aligned} - 2 x + 5 y & = - 2 \\ 4 x - 10 y & = 4 \end{aligned}$
$\begin{align*}-2x+5y &= -2\\ 4x-10y &= 4\end{align*}$
::- 2x+5y24x-10y=4

We can solve for $\begin{align*}x\end{align*}$ in the first equation as follows:
::我们可以解决第一个方程式中的 x 问题如下:

$\begin{aligned} - 2 x & = - 5 y - 2 \\ x & = \frac{5}{2} y + 1 \end{aligned}$
$\begin{align*}-2x &= -5y-2\\ x &= \frac{5}{2}y +1\end{align*}$
::-2x=52y+1 -2x=52y+1

Now, substitute this expression into the second equation and solve for $\begin{align*}y\end{align*}$ :
::现在,将此表达式替换为第二个方程, 并解决 y:

$\begin{aligned} 4 (\frac{5}{2} y + 1) - 10 y & = 4 \\ 10 y + 4 - 10 y & = 4 \\ 4 & = 4 \\ (0 & = 0) \end{aligned}$
$\begin{align*}4\left( \frac{5}{2} y+1 \right)-10y &=4\\ 10y +4 -10y &= 4\\ 4 &=4\\ (0 &= 0)\end{align*}$
::4(52y+1)-10y=410y+4-10y=44=4(0=0)

In the process of solving for $\begin{align*}y\end{align*}$ , the variable is cancelled out and we are left with only constants. We can stop at the step where 4 = 4 or continue and subtract 4 on each side to get 0 = 0. Either way, this is a true statement. As a result, we can conclude that this system has an infinite number of solutions. The lines are coincident and the system is consistent and dependent .
::在解决 y 的过程中, 变量被取消, 我们只剩下常数。我们可以停留在 4 = 4 或 4 = 4 或继续, 然后在每侧减 4 以获得 0 = 0 = 0 的阶梯上。无论如何, 这是一个真实的语句。因此, 我们可以得出这样的结论: 这个系统有无限数量的解决方案。线条是同步的, 系统是一致和依赖的。

Let's solve the following systems.
::让我们解决以下系统。

$\begin{aligned} 2 x - 3 y & = 8 \\ 6 x - 9 y & = 24 \end{aligned}$
$\begin{align*}2x-3y&=8 \\ 6x-9y&=24\end{align*}$
::2x-3y=86x-9y=24

Before we begin, first, notice that the second equation is a multiple of the first. Each term is multiplied by 3. Therefore , we know that they are the same equation and will coincide. This system has infinitely many solutions.
::在我们开始之前, 首先, 请注意第二个方程是第一个方程的倍数。每个术语乘以 3 。因此, 我们知道它们是相同的方程, 并且会同时出现。这个系统有无限多的解决方案。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find how many texts Marcel and Paran each sent.
::早些时候,你被要求寻找多少文本马塞尔和巴拉恩每人发送。

The system of linear equations represented by this situation is:
::以这种情况为代表的线性方程式系统是:

$\begin{aligned} 25 + 0.25 x & = y \\ 100 + x & = 4 y \end{aligned}$
$\begin{align*}25 + 0.25x &= y \\ 100 + x &= 4y\end{align*}$
::25+0.0.25x=y100+x=4y

Using substitution, we get:
::使用替代,我们得到:

$\begin{aligned} 100 + x & = 4 (25 + 0.25) \\ 100 + x = 100 + x \\ 0 = 0 \end{aligned}$
$\begin{align*}100 + x &= 4(25 + 0.25) \\ 100 + x = 100 + x \\ 0 = 0 \end{align*}$
::100+x=4(25+0.25)100+x=100+x+x0=0

There are an infinite number of solutions, so it can't be determined exactly how many texts Paran and Marcel sent.
::有无数的解决方案所以无法确定帕拉恩和马塞尔到底寄了多少文本

Example 2
::例2

Solve the following systems using substitution. If there is no unique solution, state whether there is no solution or infinitely many solutions.
::解决以下使用替代的系统。如果没有独有的解决办法, 请说明是否没有解决办法, 或有多少解决办法。

$\begin{aligned} y & = \frac{2}{5} x - 3 \\ 2 x - 5 y & = 15 \end{aligned}$
$\begin{align*}y &= \frac{2}{5}x-3\\ 2x-5y &=15\end{align*}$
::y= 25x- 32x- 5y= 15

Substitute the first equation into the second and solve for $\begin{align*}x\end{align*}$ :
::将第一个方程式替换为第二个方程式, 并解决 x:

$\begin{aligned} 2 x - 5 (\frac{2}{5} x - 3) & = 15 \\ 2 x - 2 x + 15 & = 15 \\ 15 & = 15 \\ (0 & = 0) \end{aligned}$
$\begin{align*}2x-5\left(\frac{2}{5}x-3\right) &= 15\\ 2x-2x+15 &= 15\\ 15 &=15\\ (0 &= 0)\end{align*}$
::2x-5(25x-3)=152x-2x+15=1515=1515=15(0=0)

Since the result is a true equation, the system has infinitely many solutions.
::由于结果是一个真正的方程式,这个系统有无数的解决方案。

Example 3
::例3

Solve the following systems using substitution. If there is no unique solution, state whether there is no solution or infinitely many solutions.
::解决以下使用替代的系统。如果没有独有的解决办法, 请说明是否没有解决办法, 或有多少解决办法。

$\begin{aligned} - x + 7 y & = 5 \\ 3 x - 21 y & = - 5 \end{aligned}$
$\begin{align*}-x+7y &= 5\\ 3x-21y &= -5\end{align*}$
::-x+7y=53x-21y=5

Solve the first equation for $\begin{align*}x\end{align*}$ to get: $\begin{align*}x=7y-5\end{align*}$ . Now, substitute this into the second equation to solve for $\begin{align*}y\end{align*}$ :
::解析 x 获得: x=7y-5 的第一个方程式。现在, 将它替换为第二个方程式。

$\begin{aligned} 3 (7 y - 5) - 21 y & = - 5 \\ 21 y - 15 - 21 y & = - 5 \\ - 15 & \neq - 5 \end{aligned}$
$\begin{align*}3(7y-5)-21y &= -5\\ 21y-15-21y &= -5\\ -15 &\ne -5\end{align*}$
::3(7-5)-21y_521y-15-21y_5

Since the result is an untrue equation, the system has no solution.
::由于结果是一个不真实的方程式,该系统没有解决办法。

Example 4
::例4

Solve the following systems using substitution. If there is no unique solution, state whether there is no solution or infinitely many solutions.
::解决以下使用替代的系统。如果没有独有的解决办法, 请说明是否没有解决办法, 或有多少解决办法。

$\begin{aligned} 3 x - 5 y & = 0 \\ - 2 x + 6 y & = 0 \end{aligned}$
$\begin{align*}3x-5y &= 0\\ -2x+6y &=0\end{align*}$
::3x-5y=0-2x+6y=0

Solving the second equation for $\begin{align*}x\end{align*}$ we get: $\begin{align*}x=3y\end{align*}$ . Now, we can substitute this into the first equation to solve for $\begin{align*}y\end{align*}$ :
::正在解决 x 的第二个方程式, 我们得到 : x= 3y 。现在, 我们可以将它替换为第一个方程式, 解决 y :

$\begin{aligned} 3 (3 y) - 5 y & = 0 \\ 9 y - 5 y & = 0 \\ 4 y & = 0 \\ y & = 0 \end{aligned}$
$\begin{align*}3(3y)-5y &=0\\ 9y -5y &= 0\\ 4y &= 0\\ y &= 0\end{align*}$
::3(3y)-5y=09y-5y=04y=0y=0

Now we can use this value of $\begin{align*}y\end{align*}$ to find $\begin{align*}x\end{align*}$ :
::现在我们可以使用 y 的这个值来查找 x:

$\begin{aligned} x & = 3 y \\ x & = 3 (0) \\ x & = 0 \end{aligned}$
$\begin{align*}x &= 3y\\ x &=3(0)\\ x &=0\end{align*}$
::x=3yx=3( 0)x=0

Therefore, this system has a solution at (0, 0). After solving systems that result in 0 = 0, it is easy to get confused by a result with zeros for the variables. It is perfectly okay for the intersection of two lines to occur at (0, 0).
::因此,这个系统在0,0时有一个解决方案。在解决了导致0=0的系统后,很容易被结果与变量的零混为一谈。在0,0时发生两行的交叉点是完全可以的。

Review
::回顾

Solve the following systems using substitution.
::使用替代方法解决以下系统。

.

$\begin{aligned} 17 x - 3 y & = 5 \\ y & = 3 x + 1 \end{aligned}$
$\begin{align*}17x - 3y &= 5\\ y &= 3x+1\end{align*}$

.

$\begin{aligned} 4 x - 14 y & = 21 \\ y & = \frac{2}{7} x + 7 \end{aligned}$
$\begin{align*}4x - 14y &= 21\\ y &= \frac{2}{7}x+7\end{align*}$

.

$\begin{aligned} - 24 x + 9 y & = 12 \\ 8 x - 3 y & = - 4 \end{aligned}$
$\begin{align*}-24x + 9y &= 12\\ 8x-3y &= -4\end{align*}$

.

$\begin{aligned} y & = - \frac{3}{4} x + 9 \\ 6 x + 8 y & = 72 \end{aligned}$
$\begin{align*}y &= - \frac{3}{4}x+9\\ 6x + 8y &= 72\end{align*}$

.

$\begin{aligned} 2 x + 7 y & = 12 \\ y & = - \frac{2}{3} x + 4 \end{aligned}$
$\begin{align*}2x + 7y &= 12\\ y &= - \frac{2}{3}x+4\end{align*}$

.

$\begin{aligned} 2 x & = - 6 y + 11 \\ y & = - \frac{1}{3} x + 7 \end{aligned}$
$\begin{align*}2x &= -6y +11\\ y &= - \frac{1}{3}x+7\end{align*}$

.

$\begin{aligned} \frac{1}{2} x - \frac{4}{5} y & = 8 \\ 5 x - 8 y & = 50 \end{aligned}$
$\begin{align*}\frac{1}{2} x - \frac{4}{5} y &= 8\\ 5x -8y &= 50\end{align*}$

.

$\begin{aligned} - 6 x + 16 y & = 38 \\ x & = \frac{8}{3} y - \frac{19}{3} \end{aligned}$
$\begin{align*}-6x + 16 y &= 38\\ x &= \frac{8}{3}y - \frac{19}{3}\end{align*}$

.

$\begin{aligned} x & = y \\ 5 x + 3 y & = 0 \end{aligned}$
$\begin{align*}x &= y\\ 5x +3y &= 0\end{align*}$

.

$\begin{aligned} \frac{1}{2} x + 3 y & = - 15 \\ y & = x - 5 \end{aligned}$
$\begin{align*}\frac{1}{2}x +3y &= -15\\ y &= x-5\end{align*}$

.

$\begin{aligned} \frac{2}{3} x + \frac{1}{6} y & = 2 \\ y & = - 4 x + 12 \end{aligned}$
$\begin{align*}\frac{2}{3}x +\frac{1}{6}y &= 2\\ y &= -4x+12\end{align*}$

.

$\begin{aligned} 16 x - 4 y & = 3 \\ y & = 4 x + 7 \end{aligned}$
$\begin{align*}16x -4y &= 3\\ y &= 4x+7\end{align*}$

.

$\begin{aligned} x - 4 y & = 10 \\ 8 y - 2 x & = - 30 \end{aligned}$
$\begin{align*}x-4y &= 10\\ 8y-2x&=-30\end{align*}$

.

$\begin{aligned} 4 x + 5 y & = 3 \\ 12 x + 15 y & = 9 \end{aligned}$
$\begin{align*}4x+5y &= 3\\ 12x+15y&=9\end{align*}$

.

$\begin{aligned} 7 x - y & = 14 \\ 35 x - 5 y & = 60 \end{aligned}$
$\begin{align*}7x-y&=14\\ 35x-5y&=60\end{align*}$
::7-y=1435x-5y=60

Review (Answers)
::回顾(答复)

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。

Section outline

General

使用替代方法的无或无限多种解决办法的解决系统

Systems with No or Infinitely Many Solutions Using Substitution ::使用替代方法的系统无或无限多的系统

Examples ::实例

Example 1 ::例1

Example 2 ::例2

y = 2 5 x − 3 2 x − 5 y = 15 \begin{align*}y &= \frac{2}{5}x-3\\ 2x-5y &=15\end{align*} ::y= 25x- 32x- 5y= 15

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Systems with No or Infinitely Many Solutions Using Substitution
::使用替代方法的系统无或无限多的系统

Examples
::实例

Example 1
::例1

Example 2
::例2

$\begin{aligned} y & = \frac{2}{5} x - 3 \\ 2 x - 5 y & = 15 \end{aligned}$
$\begin{align}y &= \frac{2}{5}x-3\\ 2x-5y &=15\end{align}$
::y= 25x- 32x- 5y= 15

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)