3.8 确定解决办法的数目-interactive

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 确定解决方案的数量

  Collapse Expand

  确定解决方案的数量

  播放段落

  Accounting For All Possibilities
  ::所有可能性会计

  

  播放段落

  Olivia is a bookkeeper for Rail Trek, a major railroad company. A bookkeeper is responsible for keeping track of a company’s business transactions. The company is offering a promotion where if you buy an adult and child ticket to a certain destination, you get a 25% discount. A child’s ticket costs $23, and the price of an adult ticket is represented by  $\begin{align*}A\end{align*}$  because it varies on different factors. Olivia is making a formula for the total discounted price of an adult and child ticket to go into a spreadsheet. Olivia claims that you can add both the adult and child ticket together and then multiply by 0.75 to find the discounted price, since 25% off is the same as 75% of the original price. She  says  the right equation is  $\begin{align*}0.75(A + 23).\end{align*}$  One of the other bookkeepers, Athena, questioned whether Olivia should find 75% of each price separately first and then add them together. Since 75% of  $\begin{align*}A\end{align*}$  is $\begin{align*}0.75A\end{align*}$ , and 75% of 23 is 17.25, she would use the formula  $\begin{align*}0.75A + 17.25.\end{align*}$  Who is right?
  ::Olivia是铁路大公司Rail Trek的记账员。 一名记账员负责跟踪公司的商业交易。 该公司正在提供一种促销, 如果你购买成人和孩子的机票到某个目的地, 就会得到25 % 的折扣。 一个孩子的票价为23美元, 成人票价由A代表, 因为它因不同因素不同而不同。 奥利维亚正在为成人和孩子的票价的折扣总额制定公式, 进入电子表格。 Olivia声称你可以将成人和孩子的票加在一起, 然后乘以0.75, 以找到折扣价格, 因为减价的25 % 等于原来的价格的75 % 。 她说正确的方程式是 0.75 ( A+23 ) 。 另外一个记账员, Athena, 质疑Olivia 是否应该先找到每个价格的75%, 然后把它们加在一起。 由于A的75%是0.75 A, 和23的75%是17.25, 她会使用0.75 A+17.25的公式。 谁是正确的?

  播放段落

  Discussion Questions
  ::讨论问题 讨论问题

  播放段落

  Which bookkeeper do you think is correct: Olivia or Athena?
  ::你认为哪个簿记员是正确的:奥利维亚还是雅典娜?

   

  ---

  播放段落

  Accounting For All Possibilities Continued
  ::所有可能性会计

  播放段落

  Before you figure out which bookkeeper made a mistake, consider that they might both be right. Perhaps both bookkeepers found two different ways to solve the same problem. This would mean that no matter what value you substituted for  $\begin{align*}A\end{align*}$  in Olivia’s formula, you would get the same result if you substituted it into Athena’s formula. To determine if the two formulas are equivalent for all numbers, you can set them equal to each other and solve the resulting multi-step equation. If the two expressions on either side of the equation are equivalent for all numbers, the variables will cancel, leaving a true statement with two constants. The true statement that you will get is a statement that a number is equal to itself. The statements 0=0 or 4=4 are both examples of true statements.
  ::在你找出哪个簿记员犯了错误之前,请考虑他们可能都是正确的。也许两个簿记员都找到了解决同一问题的两种不同方法。这意味着无论奥利维亚公式中A的值是多少,如果用雅典娜的公式取代A,你就会得到相同的结果。要确定这两个公式是否等同所有数字,你就可以将这两个公式等同起来,并解决由此产生的多步方程。如果方程两边的两个表达式都等同于所有数字,变量就会取消,留下一个真实的句子,同时保留两个常数。你将会得到的真实语句是数字本身相等的语句。语句 0=0 或 4=4是真实语句的例证。

  播放段落

  First, c heck to see if both bookkeepers are correct. If they are, then the two formulas will be equal to each other, like this:
  ::首先,检查查看两个簿记员是否正确。如果正确,那么两种公式将等同,例如:

  播放段落

  $$\begin{align*}0.75(x+23) = 0.75x+17.25\end{align*}$$
  ::0.75(x+23)=0.75x+17.25

  播放段落

  To see if this statement is true, distribute $\begin{align*}0.75 \end{align*}$  to $\begin{align*}(x + 23). \end{align*}$  Y ou get the following:
  ::要查看此语句是否真实, 请将 0. 75 分布到 (x+23) 。 您可以得到以下信息 :

  播放段落

  $$\begin{align*}0.75x+17.25 &= 0.75x+17.25\\ {\color{red}\cancel{-0.75x}} ~ ~ \cancel{0.75x}+17.25 &= \cancel{0.75x}+17.25 ~ ~ {\color{red}\cancel{-0.75x}}\\ 17.25 &= 17.25\end{align*}$$
  ::0.75x+17.25 = 0.75x+ 17.25 - 0.75x 0.75x 0.75x+ 17.25 = 0.75x+ 17.25 - 0.75x 17.25 = 17.25

  播放段落

  When you subtracted  $\begin{align*}0.75x\end{align*}$    from both sides, both $\begin{align*}x\end{align*}$ ’s canceled and you were left with the true statement $\begin{align*}17.25 = 17.25. \end{align*}$  This means that both formulas are equivalent and that both bookkeepers were correct.
  ::当您从两边减去0.75x时, 两位 x 都被取消了, 而您只剩下真实的报表17.25=17.25。 这意味着两种公式都相等, 并且两个帐务员都是正确的 。

  播放段落

  Why did this work and what does it mean?
  ::为什么这个工作,它意味着什么?

  播放段落

  With the linear equations that you have seen so far, you have only been getting one solution. However, t his isn't always the case.
  ::以你迄今所看到的线性方程来看,你只得到了一个解决方案。然而,情况并非总是如此。

  播放段落

  Use the interactive below to explore the equation   $\begin{align*}x + 3 = x + 3.\end{align*}$  
  ::使用下面的交互作用来探索公式 x+3=x+3。

   

   

   

   

  播放段落

  When you have an equation where any value substituted for the variable produces a valid solution, you  say that there are infinite solutions. In this case, any real number will make the equation true. One common use of this property is to determine whether two formulas are equivalent as  you did above.
  ::当您有一个公式, 变量中的任何值都可产生有效的解决方案时, 您会说有无限的解决方案。 在此情况下, 任何实际数字都会使方程式成为真实的 。 此属性的一个常见用途是确定两个公式是否与上面的公式相同 。

   

  ---

  播放段落

  No Solutions
  ::无解决方案

  播放段落

  Just as an equation can have infinite solutions, an equation can also have no solution. Use the interactive below to explore the equation  $\begin{align*}x = x + 1.\end{align*}$
  ::正如方程式可以有无限的解决方案, 方程式也可以没有解决方案。 使用下面的交互程序来探索公式 x=x+1 。

   

   

   

   

  播放段落

  If there are no values of $\begin{align*}x \end{align*}$  that make an equation true, we say that there are no solutions.  You will know that an equation has no solutions if the variables on both sides cancel and you are left with a false statement. The false statement that you will get is a statement that a number is equal to a number other than itself. The statements $\begin{align*}0=3 \end{align*}$  or $\begin{align*}5=4\end{align*}$  are both examples of false statements. This can occur in rate comparison problems as you saw in .
  ::如果没有 x 的值可以使等式成为真实, 我们就会说没有解决办法。 如果双方的变量都取消, 而您却被留有虚假的语句, 您就会知道方程式是没有解决办法的。 您将会得到的虚假的语句是声明一个数字等于它本身以外的数字。 语句 0=3 或 5=4 是虚假语句的两个例子。 这可能发生在您所看到的比较率问题中 。

  播放段落

  Example
  ::示例示例示例示例

  播放段落

  Sean and Madelyn are running a marathon. At the beginning of the race, Madelyn had a  $\begin{align*}\tfrac{1}{4}\end{align*}$ mile head start. Both runners are running at 12 miles per hour. How long will it take for Sean to catch Madelyn?
  ::Sean和Madelyn在跑马拉松。比赛开始时,Madelyn领先14英里。两名跑者每小时跑12英里。Sean要多久才能抓到Madelyn?

  播放段落

  You  can solve this using the following equation to determine at what point during the race Madelyn’s distance traveled will be equal to Sean’s distance traveled:
  ::您可以使用以下方程式来解答这个问题, 以决定Madelyn所走的距离与肖恩所走的距离在哪个时间点:

  播放段落

  $$\begin{align*}\text{Sean’s distance } = \text{Madelyn’s distance}\end{align*}$$
  ::Sean的距离=Madelyn的距离

  播放段落

  Sean’s distance from the start at any point in the race  can be described by the expression  $\begin{align*}12h,\end{align*}$  where  $\begin{align*}h\end{align*}$  is the number of hours he has run , because he started running at the starting line, and runs 12 miles every hour.
  ::Sean在比赛的任何一个时间点都离比赛开始的距离可以用12小时表示,h是他所跑的小时数,因为他从起点开始跑,每小时跑12英里。

  播放段落

  Madelyn’s d istance at any point in the race  can be described by the expression  $\begin{align*}12h+ \tfrac{1}{4},\end{align*}$  because she started  $\begin{align*}\tfrac{1}{4}\end{align*}$ mile  after  the starting line, and also runs 12 miles every hour ( the same speed as Sean).
  ::12h+14的表达方式可以描述马德林(Madelyn)在比赛任何时间的距离, 因为她开始在起跑线后14英里,

  播放段落

  When you substitute these distances into the equation above, you get the following equation:
  ::当您将这些距离替换为以上方程时,您可获得以下方程:

  播放段落

  $$\begin{align*}12h + \tfrac{1}{4} &= 12h \\ \\\ \tfrac{1}{4} &= 0\end{align*}$$
  ::12h+14=12h 14=0

  播放段落

  This equation has no solution because Sean will never catch Madelyn. They are running at the same speed, and she started  $\begin{align*}\tfrac{1}{4}\end{align*}$ mile ahead . Equations with no solution aren’t often used purposefully but can signal a broken equation or that two things are not equal (the distance from the starting line that Sean and Madelyn ran, in this case) .
  ::这一方程式无法解决问题,因为肖恩永远抓不到Madelyn。 他们以同样的速度运行,她领先14英里。 没有解决方案的方程式通常不会被有意使用,而是可以信号一个破碎的方程式,或者两样东西不相等(Sean和Madelyn的起点距离,在本案中是Sean和Madelyn的距离 ) 。

  播放段落

     Summary
  ::摘要

  播放段落

  An equation can have one solution, no solutions, or infinitely many solutions :
  ::等式可以有一个解决办法,没有解决办法,或无穷无尽的解决办法:

  播放段落
  • An equation with infinitely many solutions , when solved, will result in a true statement with two constants. For example,  $\begin{align*}4 = 4.\end{align*}$  
    ::有了无限多的解决方案的等式,一旦解决,将产生两个常数的真实声明。例如, 4=4。
  播放段落
  • An equation with no solutions , when solved, will result in a false statement with a number equal to another number other than itself. For example,  $\begin{align*}0 = 4.\end{align*}$  
    ::没有解决方案的等式一旦解决,将导致虚假的语句,其数字等于除它之外的其他数字。例如, 0=4。
  播放段落
  • An equation with  one solution , when solved, will result in a variable equal to a number. That number is the solution. For example,  $\begin{align*}x = 4.\end{align*}$  
    ::解答后, 包含一个解决方案的方程式将产生一个等于数字的变量。 这个数字就是解决方案。 例如, x=4 。