1.8 解决代数中的双层平方

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 解决代数中的双级平方

  Collapse Expand

  解决代数中的双级平方

  播放段落

  The Cell-U-Lar cell phone company has a data plan where 2.5 GB are included for $79.00 per month. For every 10 MB (there are 1024 MB in 1 GB) over that amount, the company charges an additional $3.00. Last month, Taylor's cell phone bill was $115. How many megabytes of data was he over the limit?
  ::手机-U-Lar手机公司有一个数据计划,其中2.5GB每月79.00美元。每10MB(每10MB(1GB中有1024MB))超过这一数额,该公司要额外收费3.00美元。 上个月,泰勒的手机账单为115美元,超过限额的数据有多少兆字节?

  播放段落

  Two-Step Equations
  ::双级平方

  播放段落

  Now, we will compound steps to solve slightly more complicated equations. We will continue to undo the operations that are in an equation . Remember that order matters. To undo the operations, work in the opposite order of the .
  ::现在,我们将组合各种步骤,以解决略微复杂的方程式。 我们将继续取消一个方程式中的操作。 记住这个顺序很重要 。 要取消操作, 工作顺序要相反 。

  播放段落

  Let's solve the following equations.
  ::让我们解决以下的方程式。

  播放段落
  1. Solve $\begin{align*}4x-9=11\end{align*}$ .
     ::解决4x9=11

  播放段落

  Notice that w e need to undo two operations. Therefore , we need to get everything on the other side of the equals sign to solve for $\begin{align*}x\end{align*}$ . First, add 9 to both sides.
  ::请注意我们需要撤销两个操作。 因此, 我们需要在等号的另一侧找到所有东西 解决 x 。 首先, 在两边加 9 。

  播放段落

  $$\begin{align*}& 4x - \bcancel{9} = 11\\ & \underline{\quad \ + \bcancel{9} \ \ +9}\\ & \quad \ \ 4x=20\end{align*}$$
  ::4-9=11+9+9+9_4x=20

  播放段落

  Now, undo the multiplication . Divide both sides by 4 to solve for $\begin{align*}x\end{align*}$ .
  ::取消乘法。 将两边除以 4 来解析 x 。

  播放段落

  $$\begin{align*}\frac{\cancel{4}x}{\cancel{4}} &= \frac{20}{4}\\ x &= 5 \end{align*}$$
  ::4x4=204x=5

  播放段落

  Check your answer: $\begin{align*}4(5) - 9 = 20 - 9 = 11\end{align*}$
  ::检查您的答案: 4(5)- 9=20- 9=11

  播放段落
  2. Solve $\begin{align*}-\frac{1}{2}g+5=14\end{align*}$ .
     ::解决 - 12g+5=14。

  播放段落

  First, subtract 5 from both sides.
  ::首先,从双方减5。

  播放段落

  $$\begin{align*}& -\frac{1}{2}g+\bcancel{5}=14\\ & \underline{\qquad \ \ -\bcancel{5} \ \ -5 \; \;}\\ & \quad \ -\frac{1}{2}g=9\end{align*}$$
  ::- 12g+5=14 - 5 - 5 - 5 - - - 12g=9

  播放段落

  Multiplying by one-half is the same as dividing by 2. To undo this fraction , multiply both sides by –2.
  ::乘以半等于除以2等于除以2。 要消除这一分数, 将两边乘以-2。

  播放段落

  $$\begin{align*} -\frac{1}{2}g &= 9\\ -\bcancel{2} \cdot -\frac{1}{\bcancel{2}}g &= 9 \cdot -2\\ g &= -18\end{align*}$$
  ::-12g=9 -2\\\\\\\\\\\\\\\\\\\\\\\\\\\\8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

  播放段落

  Check your answer: $\begin{align*}-\frac{1}{2}(-18)+5=9+5=14\end{align*}$
  ::检查您的答案 : - 12( - 18)+5=9+5=14

  播放段落
  3. Solve $\begin{align*}\frac{3}{4}x-\frac{2}{3}=\frac{7}{3}\end{align*}$ .
     ::解决34x-23=73

  播放段落

  Even though this problem involves fractions, it is still solved the same way. First, add $\begin{align*}\frac{2}{3}\end{align*}$ to both sides and then multiply both sides by the reciprocal of $\begin{align*}\frac{3}{4}\end{align*}$ .
  ::尽管这个问题涉及部分,但还是以同样的方式解决。 首先,在双方增加23个,然后以34个对等法使双方成倍地增加34个对等法。

  播放段落

  $$\begin{align*}& \frac{3}{4}x-\bcancel{\frac{2}{3}}=\frac{7}{3}\\ & \underline{\quad \ + \bcancel{\frac{2}{3}} \ +\frac{2}{3} \; \; \; \; \; \; \;}\\ & \quad \ \ \frac{3}{4}x=\frac{9}{3}=3\\ & \ \ \xcancel{\frac{4}{3} \cdot \frac{3}{4}} x=\bcancel{3} \cdot \frac{4}{\bcancel{3}}\\ & \qquad \ \ x=4\end{align*}$$
  ::34-23=73 +23 +23 +23_ 34x=93 3 43-34x=3 3-43 x=4

  播放段落

  Check your answer: $\begin{align*}\frac{3}{4}(4)-\frac{2}{3}=3-\frac{2}{3}=\frac{9}{3}-\frac{2}{3}=\frac{7}{3}\end{align*}$ .
  ::请检查您的回答: 34(4)-23=3-23=93-23=73。

  播放段落

  To summarize, for two-step equations you must do the following:
  ::简言之,对于两步方程,你必须做到如下:

  播放段落

  Step 1: Undo any addition or subtraction .
  ::第1步:取消任何增加或减法。

  播放段落

  Step 2: Undo any multiplication or division .
  ::步骤2:取消任何乘法或除法。

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Earlier, you were asked to find how many megabytes of data Taylor is over the limit.
  ::早些时候,你被要求 找到多少兆字节的数据 泰勒是超过极限。

  播放段落

  Writing an equation for the phone bill, we have: $\begin{align*}C=79+3m\end{align*}$ where C is the cost, and m is the number of 10 megabytes over the 2.5 GB. So, when we get our final answer, we need to multiply it by 10 to get the total number of megabytes over the included 2.5 GB amount. Set C = $115 and solve for m .
  ::在为电话帐单写一个方程式时,我们有:C=79+3m, C是成本所在, m是2.5GB上方的10兆字节。所以当我们得到最后答案时,我们需要乘以10, 才能在2.5GB上方获得包括2.5GB的兆字节的总数。 Set C=115, 并解决 m 。

  播放段落

  $$\begin{align*} \bcancel{79} + 3m &= 115\\ \bcancel{-79} + 3m &= -79 \\ 3m &=36\\ \frac{\cancel{3}m}{\cancel{3}} &= \frac{36}{3}\\ m &= 12\\ 12 \cdot 10 = 120 \end{align*}$$
  ::79+3m=115-79+3m793m=363m3=363m3=363m=121210=120

  播放段落

  Taylor is 120 MB over his data limit for the month.
  ::Taylor这个月的数据限额超过120 MB。

  播放段落

  Example 2
  ::例2

  播放段落

  Solve  $\begin{align*}\frac{x}{-6}-7=-11\end{align*}$ .
  ::解决 x - 6 - 7 - 7\\\ 11 。

  播放段落

  Follow Steps 1 and 2 from above. First, add 7 to both sides and then multiply both sides by –6.
  ::步骤1和步骤2从上面走下去。 首先,在双方增加7个,然后将双方乘以6。

  播放段落

  $$\begin{align*}& \ \frac{x}{-6} -\bcancel{7} = -11\\ & \underline{\qquad +\bcancel{7} \quad \ +7 \; \; \; \; \; \; \; \;}\\ & \qquad \frac{x}{-6}=-4\\ & -\bcancel{6} \cdot \frac{x}{-\bcancel{6}}=-4 \cdot -6\\ & \qquad \quad x=24\end{align*}$$
  ::x - 6 - 7 11+7+7+7_ x-6_ 4_ 6_ 6_ 6_x-6_ 6_ 6_ 6_ 4_ 6_ 4_ 6_ 6_ 4_ 6x=24

  播放段落

  Check your answer: $\begin{align*}\frac{24}{-6}-7=-4-7=-11 \end{align*}$
  ::检查您的答案: 24- 6- 7 @ @ @ @ @ @ 4- 7 @ 11

  播放段落

  Example 3
  ::例3

  播放段落

  Solve  $\begin{align*}18=6-\frac{2}{5}y\end{align*}$ . 
  ::解决18=6 -25y

  播放段落

  Again, follow Steps 1 and 2. Here, we will subtract 6 and then multiply by $\begin{align*}-\frac{5}{2}\end{align*}$ .
  ::再次遵循步骤1和2,在这里,我们将减去6,然后乘以-52。

  播放段落

  $$\begin{align*}& \qquad \quad 18=\bcancel{6}-\frac{2}{5}y\\ & \qquad \ \ \underline{-6 \ -\bcancel{6} \; \; \; \; \; \; \; \; \; \; \;}\\ & \qquad \ \ 12=-\frac{2}{5}y\\ & -\frac{5}{2} \cdot 12=-\frac{\bcancel{2}}{\cancel{5}}y \cdot - \frac{\cancel{5}}{\bcancel{2}}\\ & \quad \ -30=y\end{align*}$$
  ::18=6-25y-6-6-6-6_1225y-521225y_5225y52-30=y

  播放段落

  Check your answer: $\begin{align*}6-\frac{2}{5}(-30)=6+12=18 \end{align*}$
  ::检查您的答案: 6- 25(- 30)=6+12=18

  播放段落

  Review
  ::回顾

  播放段落

  Solve the following equations and check your answer. Reduce any fractions.
  ::解决以下方程式并检查您的答案。 减少任何分数 。

  播放段落
  1. $\begin{align*}2x-5=-17\end{align*}$
     ::2 - 517
  播放段落
  2. $\begin{align*}-4x+3=-5\end{align*}$
     ::- 4x+3+5
  播放段落
  3. $\begin{align*}-1=\frac{x}{2}-6\end{align*}$
     ::--1=x2-6
  播放段落
  4. $\begin{align*}\frac{1}{3}x+11=-2\end{align*}$
     ::13x+11=2
  播放段落
  5. $\begin{align*}\frac{3}{4}=\frac{1}{2}-\frac{1}{8}x\end{align*}$
     ::34=12-18x
  播放段落
  6. $\begin{align*}-18=\frac{x}{-5}-3\end{align*}$
     ::-18=x-5-3
  播放段落
  7. $\begin{align*}\frac{5}{6}x+4=29\end{align*}$
     ::56x+4=29
  播放段落
  8. $\begin{align*}-11x+4=125\end{align*}$
     ::- 11x+4=125
  播放段落
  9. $\begin{align*}6-x=-22\end{align*}$
     ::6-x22
  播放段落
  10. $\begin{align*}\frac{2}{7}x+8=20\end{align*}$
      ::27x+8=20
  播放段落
  11. $\begin{align*}\frac{4}{5}=-\frac{2}{5}+\frac{3}{2}x\end{align*}$
      ::4525+32x
  播放段落
  12. $\begin{align*}15-\frac{x}{-9}=21\end{align*}$
      ::15-x-9=21
  播放段落
  13. $\begin{align*}1.4x-5.6=2.8\end{align*}$
      ::1.4x-5.6=2.8
  播放段落
  14. $\begin{align*}14.4=-2.7x - 1.8\end{align*}$
      ::14.42.7x-1.8
  播放段落
  15. $\begin{align*}-\frac{5}{4}=\frac{1}{6}x+\frac{3}{12}\end{align*}$
      ::-54=16x+312

  播放段落

  When dealing with fractions, one way to “get rid” of them is to multiply everything in the equation by the Least Common Denominator (LCD) for the entire equation. Try this technique with the following equations. We will get you started in #16.
  ::当处理分数时,“除去”分数的一个方法就是将方程式中的所有方程式都由最小常见分数(LCD)乘以整个方程。尝试用以下方程式来应用这一技术。我们会在 #16 中启动。

  播放段落
  16. $\begin{align*}\frac{3}{8}x-\frac{2}{5}=\frac{7}{4}\end{align*}$ The LCD of 8, 5, and 4 is 40. $\begin{align*}40 \left(\frac{3}{8}x-\frac{2}{5}=\frac{7}{4}\right)\end{align*}$ . Distribute the 40 and solve.
      ::38-25=74 8、5和4的LCD为40 40(38x-25=74),40的LCD为40(38x-25=74)。
  播放段落
  17. $\begin{align*}\frac{10}{3}x+\frac{3}{4}=\frac{5}{2}\end{align*}$
      ::103x+34=52
  播放段落
  18. $\begin{align*}\frac{9}{10}x-\frac{1}{2}=\frac{2}{3}\end{align*}$
      ::910x-12=23
  播放段落
  19. Solve #15 using the LCD method.
      ::使用LCD方法解决15号
  播放段落
  20. Which method is easier for you to solve equations with fractions?
      ::用分数解析方程式比较容易使用哪种方法?

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  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 。