3.7 多级

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  We’ve seen that when we solve for an unknown variable , it can take just one or two steps to get the terms in the right places. Now we’ll look at solving equations that take several steps to isolate the unknown variable. Such equations are referred to as multi-step equations .
  ::我们已经看到,当我们解决一个未知变量时,只要一两步就能把条件放在正确的地方。 现在我们将研究如何解决那些采取若干步骤孤立未知变量的方程式。 这些方程式被称为多步方程式。

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  In this section, we’ll simply be combining the steps we already know how to do. Our goal is to end up with all the constants on one side of the equation and all the variables on the other side. We’ll do this by collecting like terms . Don’t forget, like terms have the same combination of variables in them.
  ::在本节中,我们只是将我们已经知道的步骤结合起来。 我们的目标是最终用方程的一边的所有常数和另一边的所有变数来完成。 我们会通过收集类似的术语来做到这一点。 不要忘记, 术语与变数的组合是一样的。

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  Solving for Unknown Values 
  ::解决未知值

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  Solve $\begin{align*}\frac{3x + 4}{3} - 5x = 6\end{align*}$ .
  ::解决 3x+43-5x=6

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  Before we can combine the variable terms, we need to get rid of that fraction .
  ::在我们把变数条件结合起来之前 我们需要摆脱这个分数

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  First let’s put all the terms on the left over a common denominator of three: $\begin{align*}\frac{3x + 4}{3} - \frac{15x}{3} = 6.\end{align*}$
  ::首先,让我们把左侧所有条件置于三个共同标准之上:3x+43-15x3=6。

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  Combining the fractions then gives us $\begin{align*}\frac{3x + 4 - 15x}{3} = 6.\end{align*}$
  ::组合分数后, 3x+4 - 15x3=6 给我们带来 3x+4 - 15x3=6 。

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  Combining like terms in the numerator gives us $\begin{align*}\frac{4 - 12x}{3} = 6.\end{align*}$
  ::分子中类似条件的组合给了我们 4 - 12x3=6 。

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  Multiplying both sides by 3 gives us $\begin{align*}4 - 12x = 18.\end{align*}$
  ::将双方乘以3 显示我们4 -12x=18

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  Subtracting 4 from both sides gives us $\begin{align*}-12x = 14.\end{align*}$
  ::从两边减四分,我们得到 - 12x=14。

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  And finally, dividing both sides by -12 gives us $\begin{align*}x = -\frac{14}{12}\end{align*}$ , which reduces to $\begin{align*}x = -\frac{7}{6}\end{align*}$ .
  ::最后,将两边除以 -12 给我们x1412, 减为x76。

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  Solving Multi-Step Equations Using the Distributive Property
  ::使用分配财产解决多股

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  You may have noticed that when one side of the equation is multiplied by a constant term , we can either distribute it or just divide it out. If we can divide it out without getting awkward fractions as a result, then that’s usually the better choice, because it gives us smaller numbers to work with. But if dividing would result in messy fractions, then it’s usually better to distribute the constant and go from there.
  ::你可能已经注意到,当方程式的一方乘以一个常数时,我们要么分配它,要么将其分割开来。 如果我们可以在不因此获得尴尬的分数的情况下将其分割出去,那么这通常是更好的选择,因为它给了我们较少的数量来工作。 但如果分数会导致混乱的分数,那么通常比较好的做法是分配常数并从中取而代之。

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  Using the Distributive Property 
  ::使用分配财产

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  1. Solve $\begin{align*}7(2x - 5) = 21\end{align*}$ .
  ::1. 解决7(2x-5)=21。

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  The first thing we want to do here is get rid of the " data-term="Parentheses" role="term" tabindex="0"> parentheses . We could use the , but it just so happens that 7 divides evenly into 21. That suggests that dividing both sides by 7 is the easiest way to solve this problem.
  ::我们首先要做的就是把括号扔掉。我们可以使用括号,但7个平分到21个,这说明将双方除以7是解决这一问题的最容易的方法。

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  If we do that, we get $\begin{align*}2x - 5 = \frac{21}{7} \end{align*}$ or just $\begin{align*}2x - 5 = 3\end{align*}$ . Then all we need to do is add 5 to both sides to get $\begin{align*}2x = 8\end{align*}$ , and then divide by 2 to get $\begin{align*}x = 4\end{align*}$ .
  ::如果我们这样做,我们就会得到2x-5=217或只有2x-5=3。 然后我们只需要在两边加5,再加2x=8,再除以2再取x=4。

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  2. Solve $\begin{align*}17(3x + 4) = 7\end{align*}$ .
  ::2. 解决17(3x+4)=7。

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  Once again, we want to get rid of those parentheses. We could divide both sides by 17, but that would give us an inconvenient fraction on the right-hand side. In this case, distributing is the easier way to go.
  ::再一次,我们想摆脱这些括号。 我们可以把两边除以17, 但右侧会给我们一个不方便的部分。 在这种情况下,分配比较容易。

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  Distributing the 17 gives us $\begin{align*} 51x + 68 = 7\end{align*}$ . Then we subtract 68 from both sides to get $\begin{align*}51x = -61\end{align*}$ , and then we divide by 51 to get $\begin{align*}x = \frac{-61}{51}\end{align*}$ . (Yes, that’s a messy fraction too, but since it’s our final answer and we don’t have to do anything else with it, we don’t really care how messy it is.)
  ::分配17分之51x+68=7。 然后,我们从双方减去68分之68以获得51xQQQQ*61,然后再除以51以获得xQQQ6151。 (是的,这也是一个混乱的部分,但既然这是我们的最后答案,而且我们不必做其他事情,我们真的不在乎它有多乱。 )

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  3. Solve $\begin{align*}4(3x - 4) - 7(2x + 3) = 3\end{align*}$ .
  ::3. 解决4(3x-4)-7(2x+3)=3。

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  Before we can collect like terms, we need to get rid of the parentheses using the Distributive Property. That gives us $\begin{align*}12x - 16 - 14x - 21 = 3\end{align*}$ , which we can rewrite as $\begin{align*}(12x - 14x) + (-16 - 21) = 3\end{align*}$ . This in turn simplifies to $\begin{align*}-2x - 37 = 3\end{align*}$ .
  ::在我们收集类似术语之前,我们需要用分配财产来去除括号。 这使得我们12x-16- 14x- 21=3, 我们可以改写为( 12x-14x)+( -16- 21)=3。 这反过来又简化为-2x-37=3。

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  Next we add 37 to both sides to get $\begin{align*}-2x = 40\end{align*}$ .
  ::接下来,我们向双方增加37, 以获得 - 2x=40。

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  And finally, we divide both sides by -2 to get $\begin{align*}x = -20\end{align*}$ .
  ::最后,我们把两边除以 -2 来得到x20。

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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 following equation for $\begin{align*}x\end{align*}$ : $\begin{align*}0.1(3.2 + 2x) + \frac{1}{2} \left ( 3 - \frac{x}{5} \right ) = 0\end{align*}$
  ::为 x 解决以下方程式: 0.1(3.2+2x)+12(3-x5)=0

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  This function contains both fractions and decimals. We should convert all terms to one or the other. It’s often easier to convert decimals to fractions, but in this equation the fractions are easy to convert to decimals—and with decimals we don’t need to find a common denominator!
  ::此函数包含分数和小数。 我们应该将所有术语转换为一个或另一个。 将小数转换为分数通常比较容易, 但在此方程式中, 分数很容易转换为小数, 而以小数计算, 我们不需要找到共同的分母 。

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  In decimal form, our equation becomes $\begin{align*}0.1(3.2 + 2x) + 0.5(3 - 0.2x) = 0\end{align*}$ .
  ::以小数表示,我们的方程式为 0.1(3.2+2x)+0.5(3-0.2x)=0。

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  Distributing to get rid of the parentheses, we get $\begin{align*}0.32 + 0.2x + 1.5 - 0.1x = 0\end{align*}$ .
  ::为删除括号,我们分配0.32+0.2x+1.5-0.1x=0。

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  Collecting and combining like terms gives us $\begin{align*}0.1x +1.82 = 0\end{align*}$ .
  ::收集并合并类似术语给我们0. 1x+1.82=0 。

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  Then we can subtract 1.82 from both sides to get $\begin{align*}0.1x = -1.82\end{align*}$ , and finally divide by 0.1 (or multiply by 10) to get $\begin{align*}x = -18.2\end{align*}$ .
  ::然后我们可以从双方中减去1.82 以获得0.1x1.82,最后除以0.1(或乘以10)以获得 x18.2。

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  Review 
  ::回顾

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  Solve the following equations for the unknown variable.
  ::为未知变量解决以下方程式。

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  1. $\begin{align*}3(x - 1) - 2(x + 3) = 0\end{align*}$
     ::3(x--1)-2(x+3)=0
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  2. $\begin{align*}3(x + 3) - 2(x - 1) = 0\end{align*}$
     ::3(xx+3)-2(x-1)=0
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  3. $\begin{align*}7(w + 20) - w = 5\end{align*}$
     ::7(w+20)-w=5
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  4. $\begin{align*}5(w + 20) - 10w = 5\end{align*}$
     ::5(w+20)-10w=5
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  5. $\begin{align*}9(x - 2) - 3x = 3\end{align*}$
     ::9(x-2)-3x=3
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  6. $\begin{align*}12(t - 5) + 5 = 0\end{align*}$
     ::12(t-5)+5=0
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  7. $\begin{align*}2(2d + 1) = \frac{2}{3}\end{align*}$
     ::2(2d+1)=23
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  8. $\begin{align*}2 \left ( 5a - \frac{1}{3} \right ) = \frac{2}{7}\end{align*}$
     ::2(5a)-(13)=27
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  9. $\begin{align*}\frac{2}{9} \left ( i + \frac{2}{3} \right ) = \frac{2}{5}\end{align*}$
     ::29(i+23)=25
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  10. $\begin{align*}4 \left ( v + \frac{1}{4} \right ) = \frac{35}{2}\end{align*}$
      ::4(v+14)=352
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  11. $\begin{align*}\frac{g}{10} = \frac{6}{3} \end{align*}$
      ::g10=63
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  12. $\begin{align*}\frac{m + 3}{2} - \frac{m}{4} = \frac{1}{3} \end{align*}$
      ::m+32-m4=13
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  13. $\begin{align*}5 \left ( \frac{k}{3} + 2 \right ) = \frac{32}{3}\end{align*}$
      ::5(k3+2)=323
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  14. $\begin{align*}\frac{3}{z} = \frac{2}{5}\end{align*}$
      ::3z=25
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  15. $\begin{align*}\frac{2}{r} + 2 = \frac{10}{3} \end{align*}$
      ::2r+2=103
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  16. $\begin{align*}\frac{12}{5} = \frac{3 + z}{z}\end{align*}$
      ::125=3+zz

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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 。