解析多级

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Your friend is in a band and asks you to help out by selling tickets to the next show. In a rush, they hand you money from two other people and excitedly tell you about how many tickets were sold. Before you can ask about the price of the tickets, they leave. How much money should you charge for a ticket? To answer this question we will have to perform several operations .
::你的朋友在乐队里, 要求你帮忙, 卖下一场演出的票。 匆忙的时候, 他们把另外两人的钱交给你, 并兴奋地告诉你售出几张票。 在你问票价之前, 他们就离开了。 您要花多少钱买一张票? 为了回答这个问题, 我们不得不进行几次操作 。

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Solving Multi-Step Equations
::解析多级

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The types of equations in this concept involve at least three steps and  are referred to as multi-step equations . We will cover what to do if there are multiple terms with the variable in the equation , and what to do if the equation involves " data-term="Parentheses" role="term" tabindex="0"> parentheses .  
::这一概念中的方程式类型至少涉及三个步骤,被称为多步骤方程式。我们将涵盖如果方程式中变量存在多个条件,以及如果方程式包含括号,我们该怎么办。

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Combining Like Terms
::将类似术语合并

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You will encounter equations that have multiple terms with the variable in the equation or multiple terms that are just numbers or constant terms. We can solve equations like this if we combine the like terms . Recall, like terms are expressions that have the same variables and each variable is raised to the same power. We can combine like terms by performing an operation between the coefficients in these expressions. 
::您将会遇到方程中与变量有多个条件的方程式, 或公式中与变量有多个条件的方程式, 或只是数字或不变条件的多个条件。 如果我们将类似条件组合在一起, 我们可以解决这样的方程式 。 回顾, 类似术语的表达式具有相同的变量, 每个变量被提升到相同的力量 。 我们可以通过在这些表达式中的系数之间执行操作, 将类似术语组合在一起 。

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For example, 
::例如,

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$$\begin{align*}4xyz+13xyz=(4+13)xyz=17xyz\end{align*}$$
::4xyz+13xyz=(4+13)xyz=17xyz

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When you have an equation with like terms, the first step is to combine them. Let's see how this works in the following example.
::当你有一个类似条件的方程式时, 第一步是将它们结合起来。 让我们在下面的例子中看看它是如何运作的 。

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Example 1
::例1

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Solve $\begin{align*}8x-17=4x+23 \end{align*}$ .
::解决 8x - 17=4x+23。

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Solution: This equation has $\begin{align*}x\end{align*}$ on both sides of the equal sign. Therefore , we need to move one of the $\begin{align*}x\end{align*}$ -terms to the other side of the equation. It does not matter which $\begin{align*}x\end{align*}$ - term you move. We will move the $\begin{align*}4x\end{align*}$ to the other side so that, when combined, the coefficient of the  $\begin{align*}x\end{align*}$ -term is positive.
::解决方案 : 此方程式在等号的两侧都有 x 。 因此, 我们需要将一个 x 条件移到方程式的另一侧 。 您移动哪个 x 条件并不重要 。 我们将将 4x 移到另一侧, 这样, 如果加起来, x 条件的系数是正数 。

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$$\begin{align*}& \; \;\; 8x-17=\cancel{4x}+23\\ & \underline{-4x \quad \ \ -\cancel{4x} \; \; \; \; \; \; \; \;} && \color{gray}{\text{combine like terms} \ 8x \ \text{and} \ 4x}\\ & 4x-\cancel{17}=23\\ & \underline{\quad \ +\cancel{17} +17 }\\ & \qquad \frac{\cancel{4}x}{\cancel{4}}=\frac{40}{4}\\ & \qquad \ \ x=10\end{align*}$$
::8-17=4x+23-4x-4x-4x-4x-combine 类似术语 8x 和 4x4x-17=23 +17+17_17_4x4=404 x=10

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Check your answer:
::检查您的答案 :

$$\begin{align*}8(10)-17 &= 4(10)+23\\ 80-17 &= 40+23 \\ 63 &= 63\end{align*}$$

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Example 2
::例2

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Suppose the sum of two consecutive integers is 59, What are the integers? 
::假设两个连续整数的总和是59, 整数是多少?

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Solution: First, it is important to understand what the word consecutive means. Consecutive means next to; two integers that are consecutive are integers that follow each other in order. For example, 3 and 4 are consecutive integers because in an ordered list of integers 3 and 4 are next to each other, or 4 follows 3.
::解决方案 : 首先, 重要的是要理解连续单词的含义 。 连续单词的意思是 。 连续单词的意思是 。 连续单词的意思是 : 两个连续的整数是 顺序相依的整数 。 例如, 3 和 4 是 连续单数 3 和 4 , 因为定单的整数 3 和 4 相邻, 或者 4 相随于 3 。

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To describe these unknown integers with variables, notice that the difference between them is 1. If we let the first integer be  $\begin{align*}n\end{align*}$ , then the next consecutive integer would be  $\begin{align*}n+1.\end{align*}$
::要用变量描述这些未知的整数, 请注意, 它们之间的差为 1. 如果我们让第一个整数为 n, 那么下一个连续的整数将是 n+ 1 。

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According to the question, the sum of  $\begin{align*}n\end{align*}$  and  $\begin{align*}n+1\end{align*}$  is 59 or  $\begin{align*}n+n+1=59\end{align*}$  .
::根据问题,n和n+1的总和为59或n+n+1=59。

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$$\begin{align*}n+n+1=59 \\ 2n+\cancel{1}=59\\ \underline{\qquad \ -\cancel{1} \ -1} \\ 2n=58 \\ \frac{\cancel{2}n}{\cancel{2}}=\frac{58}{2} \\ n=29\end{align*}$$
::n+n+1=592n+1=59-1=59-1-1-1_2n=582n2=582n=29

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If  $\begin{align*}n=29\end{align*}$ , then  $\begin{align*}n+1=29+1=30. \end{align*}$
::如果 n=29,则n+1=29+1=30。

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Checking our answer:  $\begin{align*}29+29+1=59 \ \text{or} \ 29+30=59.\end{align*}$
::查询我们的答案: 29+29+1=59 或 29+30=59。

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by John Travis explains how to solve equations when combining like terms is necessary.    
::John Travis解释在需要将类似条件合并时如何解析方程式。

 

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Example 3
::例3

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Your friend hands you a pile of tickets to sell and $492. They excitedly tell you that Sofia sold 23 tickets and Ridwan sold 18 tickets. Before you can ask about the price of the tickets, they leave. How much money should you charge for a ticket?
::你的朋友给了你一大笔售票和492美元,他们激动地告诉你,索菲亚卖了23张票,里德万卖了18张票。在问票价之前,他们离开了。你应该收取多少钱买票?

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Solution: Let t be the price of a ticket. Sofia sells $\begin{align*}23t\end{align*}$  worth of tickets and Ridwan sells $\begin{align*}18t\end{align*}$  worth of tickets. Together, that is  
::解决:让我们来决定一张票的价格吧。索菲亚卖了23张票,里德万卖了18张票。加在一起,这就是说,索菲亚卖了23张票,里德万卖了18张票。

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$$\begin{align*}23t+18t&=492 \\ \frac{\cancel{41}t}{\cancel{41}}&=\frac{492}{41} \\ t&=12\end{align*}$$
::23t+18t=49241t41=49241t=12

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You should charge $12 per ticket.
::每张票要收12美元

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Distributive Property
::分配财产

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Another type of equation with multiple steps is an equation with parentheses. To solve  these equations, first we will use the . Recall the distributive property states 
::另一种具有多个步骤的方程是带有括号的方程。要解答这些方程,首先我们要使用 。回顾分配属性状态

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$$\begin{align*}a(b+c)=ab+ac.\end{align*}$$

::a(b+c)=ab+ac。

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Example 4
::例4

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Solve $\begin{align*}2(x-7)=10\end{align*}$ .
::解决 2(x-7)=10。

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Solution:  Let’s simplify the left side of this equation by using the distributive property.
::解决方案:让我们使用分配属性来简化这个等式的左侧。

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$$\begin{align*}2(x-7) &= 10\\ 2x-14&= 10 && \color{gray}{\text{distributive property}}\\ \end{align*}$$
::2(x-7)=102x-14=10分配财产

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Now, this is a two-step equation and we continue to solve it using the techniques we learned in the previous section.
::现在,这是一个两步方程,我们继续使用我们在上一节中学到的技术来解决它。

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$$\begin{align*}& \ 2x-\cancel{14}=10\\ & \underline{\quad \ \ + \cancel{14} \ + 14 \; \;}\\ & \qquad \frac{\cancel{2}x}{\cancel{2}}=\frac{24}{2}\\ & \qquad \ \ x=12\end{align*}$$
::2 - 14= 10+14+14+14_ 2x2=242 x=12

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Check your answer: $\begin{align*}2(12-7)=2 \times 5=10. \end{align*}$
::检查您的答案: 2( 212- 7) = 2x5= 10 。

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Example 5
::例5

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The specific heat capacity of a substance is a ratio of the heat added or removed from an object to the change in temperature by 1 degree Celsius. The equation is  $\begin{align*}Q=mC(T_F - T_I)\end{align*}$  where  $\begin{align*}Q\end{align*}$  is the heat added,  $\begin{align*}m\end{align*}$  is the mass of the substance,  $\begin{align*}C\end{align*}$  is the specific heat capacity, and  $\begin{align*}T_F\end{align*}$  and  $\begin{align*}T_I\end{align*}$  are the final and initial temperature, respectively. ¹
::物质的具体热容量是从物体上添加或去除的热量与温度变化1摄氏度之比。等式是Q是加热量,M是物质的质量,C是特定热量,TF和TI分别是最后温度和初始温度。

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We have 100 grams of water ( $\begin{align*}m=100\end{align*}$ ) that is at 20 degrees Celsius ( $\begin{align*}T_I = 20\end{align*}$ ). The water absorbs 1100 Joules of energy ( $\begin{align*}Q=1100\end{align*}$ ).  The specific heat capacity of water is 4.18 $\begin{align*}J/g ^\circ C\end{align*}$  ( $\begin{align*}C=4.18\end{align*}$ ). What is the final temperature of the water? 
::我们有100克水(m=100),温度为20摄氏度(TI=20),吸收了1100焦耳的能量(1100),水的具体热能为4.18 J/gC(C=4.18),水的最后温度是多少?

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Solution: Substituting these values into the equation, we have:
::解决方案:将这些值转换成等式,我们有:

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$$\begin{align*}&1100=100 \times 4.18(T_F - 20) \\ &1100=418(T_F - 20) \\ &1100=418 T_F - \cancel{8360} \\ &\underline{+8360 \qquad \qquad+\cancel{8360}} \\ &9460=418 T_F \\ &\frac{9460}{418} = \frac{\cancel{418} T_F}{\cancel{418}} \\ &22.63 = T_F\end{align*}$$
::1100=100x4.18(TF - 201)1100=418(TF - 201)1100=418TF - 8360+8360+8360_ 8360_ 9460=418TF9460418=418TF41822.63=TF

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The final temperature is 22.63 degrees Celsius (This is 72.73 degrees Fahrenheit).
::最后温度为22.63摄氏度(华氏72.73摄氏度)。

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Bringing These Ideas Together
::把这些想法集为一体

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Lastly, we bring these two ideas together. What do you do if there are like terms and parentheses? Since you cannot combine like terms inside and outside parentheses, you have to first do the distributive property, so that you can access the terms inside the parentheses. 
::最后,我们把这两个想法放在一起。如果有像术语和括号一样的术语和括号,你会怎么做?既然你不能像括号内和括号外的术语组合在一起,你必须首先做分配财产,这样你就可以使用括号内的条款。

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Let's see how this works in the following example. 
::让我们在下面的例子中看看这是怎么回事。

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Example 6
::例6

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Solve  $\begin{align*}0.6(2x-7)=5x-5.1\end{align*}$
::溶解 0.6( 2x- 7) = 5x- 5. 1

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Solution:  First, we distribute 0.6 to the terms inside the parentheses and then we combine like terms.
::解决办法:首先,我们分发0.6至括号内的条款,然后将类似条款合并。

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$$\begin{align*}& \ 0.6(2x+7)=4.3x-5.1\\ & \ \ 1.2x+4.2=\cancel{4.3x}-5.1 && \color{gray}{\text{distributive property}}\\ & \underline{-4.3x \qquad \ -\cancel{4.3x} \qquad} && \color{gray}{\text{combine like terms}}\\ & \ -3.1x+\cancel{4.2}=-5.1 \\ & \underline{\qquad \qquad -\cancel{4.2} \ \ \ -4.2 }\\ & \quad \quad \ \frac{-\cancel{3.1}x}{-\cancel{3.1}}=\frac{-9.3}{-3.1}\\ & \qquad \qquad \ \ x=3\end{align*}$$
::0.6(2x+7) =4.3x-5.1 1.2x+4.2=4.3x-5.1 分配属性-4.3x-4.3x-4.3x_combine,类似术语-3.1x+4.2_5.1-4.2-4.2_-3.1x-3.1x-3.1.9.3-3.1x=3

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Check your answer by substituting into the original equation and determine if both sides are equal.
::通过替换原始方程式来检查您的答案, 并确定双方是否相等 。

$$\begin{align*}0.6(2(3)+7) &= 4.3(3)-5.1\\ 0.6 \times 13 &= 12.9-5.1 \\ 7.8 &= 7.8\end{align*}$$

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by sarakarch demonstrates how to solve equations using the distributive property and by combining like terms. 
::由sarakarch 演示如何使用分配属性和合并类似术语来解析方程式。

 

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Example 7
::例7

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Solve $\begin{align*}2(3u-1)+2u=5-(2u-3)\end{align*}$ .
::解决2(2,3u-1)+2u=5-(2u-3)。

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Solution: This equation combines the two previous examples. First, use the distributive property.
::解决方案: 此方程式结合了前两个示例。 首先, 使用分配属性 。

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$$\begin{align*}2(3u-1)+2u &= 5-(2u-3)\\ 6u-2+2u &= 5-2u+3 && \color{gray}{\text{distributive property}}\end{align*}$$
::2(3u-1)+2u=5-(2u-3)6u-2+2u=5-2u+3分配性财产

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Do not forget to distribute -1 to each of the terms in the  parentheses on the right side of the equation. Now, combine like terms and solve the equation.
::不要忘记将 - 1 分布到方程式右侧括号中的每个词。 现在, 将类似条件合并并解析方程式 。

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$$\begin{align*}& \ \ \ 8u-2=8-\cancel{2u}\\ & \underline{+2u \qquad \qquad +\cancel{2u}} && \color{gray}{\text{combine like terms}}\\ & \ 10u-\cancel{2}=8 \\ & \underline{\qquad +\cancel{2} \ \ +2} \\ & \quad \ \frac{\cancel{10}u}{\cancel{10}}=\frac{10}{10}\\ & \qquad \ u=1\end{align*}$$
::8-2=8-2u+2u+2u+2u_combines like unu-2=8+2+2+2_10=1010u=1

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Check your answer by substituting into the original equation. 
::通过替换原始方程式来检查您的答案 。

$$\begin{align*}2(3(1)-1)+2(1) &= 5-(2(1)-3)\\ 2 \times 2+2 &= 5-(-1) \\ 4+2 &= 6\\ 6&=6\end{align*}$$

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Example 8
::例8

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Solve $\begin{align*}2(n-4) = 2n+5\end{align*}$ . 
::解决2(n-4)=2n+5。

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Solution:  We apply the distributive property first:  $\begin{align*}2n-8=2n+5\end{align*}$ . Next, we combine like terms on opposite sides of the equation.
::解决方案:我们首先应用分配属性: 2n-8=2n+5。 下一步,我们在等式的对面合并类似条件。

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$$\begin{align*}\ \ 2n - 8 = 2n + 5 \\ \underline{-2n \quad \ \ -2n \qquad} \\ -8=5 \qquad\end{align*}$$
::2n-8=2n+5-2n-2n-2n_2n_2n_8=5

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The terms with the variable, the $\begin{align*}2n\end{align*}$ , canceled each other out. Since -8=5 is not a true statement, there is no solution.
::变量的条件 2n , 彼此取消 。 由于 - 8=5 不是一个真实的语句, 因此没有解决方案 。

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Example 9
::例9

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Solve  $\begin{align*}-3(m-6)=-3m+18\end{align*}$ .   
::解决 - 3(m-6)\\% 3m+18。

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Solution: Use the distributive property to remove the parentheses: $\begin{align*}-3m+18 = -3m+18\end{align*}$ . Combining the  m -terms on both sides, we have
::解决方案:使用分配属性去掉括号: - 3m+183m+18。将双方的 m 条件合并在一起,我们发现

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$$\begin{align*}\ \ -3m+18=-3m+18 \\ \underline{+3m \quad \qquad +3m \qquad} \\ 18=18 \qquad \quad\end{align*}$$
::-3m+183m+18+3m+3m+3m_18=18

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The terms with the variables, the $\begin{align*}3m\end{align*}$ , canceled each other out. Since 18=18 is a true statement and the value of $\begin{align*}m\end{align*}$  does not matter, the solutions are all real numbers.  
::变量的条件, 3m, 相互取消 。 由于 18 = 18 是真实的语句, m 的价值并不重要, 解决方案都是真实的数字 。

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Summary
::摘要

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To solve linear equations of one variable:
::要解析一个变量的线性方程式 :

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• First, use the distributive property. 
  ::首先,使用分配财产。
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• Second, combine like terms on the same side of the equation. 
  ::第二,将方程的同一侧的类似条件结合起来。
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• Third, combine like terms on opposite sides of the equation.
  ::第三,在等式的对面结合类似条件。
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• Fourth, undo the operations that are done to the variable in reverse order. This usually means you should perform addition or subtraction first and then multiplication or division. To undo an operation, perform the inverse operation.
  ::第四,以反向顺序撤销对变量的操作。这通常意味着您应该先执行加法或减法,然后进行乘法或除法。要取消操作,请执行反向操作。
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• Lastly, check your answer by substituting your solution back into the original equation.
  ::最后,检查一下你的答案 换回原来的方程

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Review
::回顾

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Solve each equation and check your solution.
::解决每一个方程 检查你的解决方案。

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1. $\begin{align*}8-9x-5=x+23\end{align*}$
::1. 8-9x-5=x+23

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2. $\begin{align*}y-\frac{4}{5}=\frac{2}{3}y+\frac{8}{15}\end{align*}$
::2. y-45=23y+815

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3.  $\begin{align*}-52.7+3.1w-6.75=1.05w\end{align*}$
::3.-52.7+3.1w-6.75=1.05w

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4.  $\begin{align*}4s-3\sqrt{6}+5\sqrt{2}=\sqrt{6}+\sqrt{2}\end{align*}$
::4. 4s-36+5262

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5.  $\begin{align*}-6(2t-5)=18\end{align*}$
::5-6(2-5)=18

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6. $\begin{align*}5\left(\frac{r}{3}+2\right)=\frac{32}{3}\end{align*}$
::6.5(r3+2)=323

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7.  $\begin{align*}-3.1(0.1p-0.4)=21.39\end{align*}$
::7.-3.1(0.1p-0.4)=21.39

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8.  $\begin{align*}\sqrt{2}(q+1)=3\sqrt{2}\end{align*}$
::8. 2(q+1)=32

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9.  $\begin{align*}3(n-4)=5(n+6)\end{align*}$
::9. 3(n-4)=5(n+6)

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10.  $\begin{align*}m-12+3m=-2(m+18)\end{align*}$
::10.-12+3m=2(m+18)

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11.  $\begin{align*}2-(3h+7)=-h+19\end{align*}$
::11.2-(3h+7)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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12.  $\begin{align*}\frac{1}{6}(d+2)=2\left(\frac{3d}{2}-\frac{5}{4}\right)\end{align*}$
::12. 16(d+2)=2(3d2-54)

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13. $\begin{align*}\frac{x+1}{2}=3\end{align*}$
::13. x+12=3 13 x+12=3

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14. $\begin{align*}\frac{1}{2}(x + 1) = 8 \end{align*}$
::14. 12(x+1)=8

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Explore More
::探索更多

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1. In an attempt to find the resistance of a new component, an engineer tests it in series with standard resistors. 
::1. 为了寻找新部件的抗力,工程师用标准抗力器进行系列测试。

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• In one test, a fixed voltage causes a 4.8 amp current in a circuit made up from the new component plus a $\begin{align*}15\Omega\end{align*}$  (Ohms) resistor in series.
  ::在一次试验中,固定电压在由新部件组成的电路中产生4.8个脉冲电流,加上一个15-(Omms)系列阻力器。
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• In another test with the same voltage, there is a 2.8 amp current when the component is placed in a series circuit with a $\begin{align*}50\Omega\end{align*}$ resistor.
  ::在用同一电压进行的另一个试验中,当部件安装在配有50-30抗力的系列电路中时,就会有2.8个电流。

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Calculate the resistance of the new component.
::计算新组件的阻力。

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(Hints: $\begin{align*}V=I \cdot R\end{align*}$  , where V is the voltage, I is the current, and R is the total resistance.  When resistors are connected in series, the total resistance is the sum of the resistors. ² Set up an equation for each test, and then note that the voltage is the same for each test.)
:提示: V=IR, V是电压, I是电流, R是总阻力。当阻力器连成序列时,总阻力是阻力器的总和。 2 为每次测试设置一个方程式, 并注意每次测试的电压相同 。)

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by CK-12 demonstrates how to solve real-world problem using multi-step equations. 
::使用 CK-12 演示如何使用多步方程解决现实世界问题。

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2. Manoj and Tamar are arguing about a number trick they heard. Tamar tells Andrew to think of a number, multiply it by five and subtract three from the result. Then Manoj tells Andrew to think of a number, add five and multiply the result by three. Andrew says that whichever way he does the trick he gets the same answer.
::2. Manoj和Tamar在争论他们听到的数个骗局。Tamar告诉Andrew想想一个数字,乘以5,从结果中减去3。然后Manoj告诉Andrew想想一个数字,加以5,再乘以3。Andrew说,不管他用什么方法做这个把戏,他都会得到同样的答案。

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a. What was the number Andrew started with?
::a. 安德鲁最初的号码是多少?

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b. What was the result Andrew got both times?
::b. Andrew两次得到的结果是什么?

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c. Name another set of steps that would have resulted in the same answer if Andrew started with the same number.
::c. 列出另一套步骤,如果安德鲁以同一数字开始,这些步骤本可得出同样的答案。

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3. Solve  $\begin{align*}\frac{3}{4}+\frac{2}{3}x=2x+\frac{5}{6}\end{align*}$  by using the LCD method introduced in the previous read. Multiply every term by the LCD of 4, 3, and 6.
::3. 使用前文中引入的液化催化分解法,解决34+23x=2x+56。 用4、3和6的液化分解法乘以每个术语。

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4. Joel is going to start a business. To cover his start-up costs he takes out two loans. 
::4. Joel打算开办企业,为了支付开办费用,他借了两笔贷款。

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• Loan 1: Joel made an initial payment of $2,000 and will make monthly payments of $500 per month.
  ::贷款1:Joel首次支付2 000美元,每月支付500美元。
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• Loan 2: Joel made an initial payment of $1,300 and will make monthly payments of $200 per month.
  ::贷款2:Joel首次支付1 300美元,每月支付200美元。

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How many months will it take Joel to pay off these two loans if they are $21,500 in total?
::如果这两笔贷款总共21 500美元,Joel要花多少个月才能偿还这笔贷款?

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Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

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Please see the Appendix. 
::请参看附录。

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PLIX
::PLIX

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Try these interactives that reinforce the concepts explored in this section:
::尝试这些强化本节所探讨概念的交互作用 :

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References
::参考参考资料

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1. “Heat Capacity,” last edited June 22, 2017, .
   ::2017年6月22日编辑。
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2. “Ohm’s Law,” last edited June 21, 2017, .
   ::2017年6月21日编辑。