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  • 平等两步平等和平等财产

    Two Step Equations and Properties of Equality 
    ::平等的两个步骤和平等的属性

    We’ve seen how to solve for an unknown by isolating it on one side of an equation and then evaluating the other side. Now we’ll see how to solve equations where the variable takes more than one step to isolate.
    ::我们看到了如何通过将未知方程式隔离在方程式的一面,然后对另一面进行评估来解决未知方程式的问题。 现在我们来看看如何解决方程式问题,让变量采取不止一步的孤立。

    Real-World Application: Marbles 
    ::真实世界应用: Marbles

    Rebecca has three bags containing the same number of marbles, plus two marbles left over. She places them on one side of a balance. Chris, who has more marbles than Rebecca, adds marbles to the other side of the balance. He finds that with 29 marbles, the scales balance. How many marbles are in each bag? Assume the bags weigh nothing.
    ::Rebecca有三个袋子,内含同样的大理石,外加两块大理石,她把它们放在一个平衡的一侧。Chris比Rebecca有更多的大理石,他把大理石加在另一个平衡的一侧。他发现有29个大理石,其中的天平是平衡的。每个袋子里有多少大理石?装在袋子里,装在袋子里没有重物。

    We know that the system balances, so the weights on each side must be equal. If we use x to represent the number of marbles in each bag, then we can see that on the left side of the scale we have three bags (each containing x marbles) plus two extra marbles, and on the right side of the scale we have 29 marbles. The balancing of the scales is similar to the balancing of the following equation.
    ::我们知道系统平衡,因此每边的重量必须相等。如果我们用x来代表每个袋子中的大理石数量,那么我们可以看到,在比例表的左边,我们有三个袋子(每个袋子含有x大理石)加上两个额外的大理石,在比例表的右边,我们有29个大理石。比例的平衡与以下方程式的平衡相似。

    3 x + 2 = 29

    ::3x+2=29

    “Three bags plus two marbles equals 29 marbles”
    ::“三个袋加两个大理石球等于29个大理石”

    To solve for x , we need to first get all the variables ( terms containing an x ) alone on one side of the equation. We’ve already got all the x ’s on one side; now we just need to isolate them.
    ::要解决 x 问题, 我们首先需要将所有变量( 包含 x 的术语) 单放在方程的一边。 我们已经把所有 x 的变量都放在一边了; 现在我们只需要孤立它们 。

    3 x + 2 = 29 3 x + 2 − 2 = 29 − 2 Get rid of the 2 on the left by subtracting it from both sides. 3 x = 27 3 x 3 = 27 3   Divide both sides by 3. x = 9

    ::3x+2=293x+2-2=29-2=29-2 将左侧的2除去,从两侧减去3x=273x3=273除以3.x=9

    There are nine marbles in each bag.
    ::每个袋子里有九颗弹珠

    We can do the same with the real objects as we did with the equation. Just as we subtracted 2 from both sides of the equals sign, we could remove two marbles from each side of the scale. Because we removed the same number of marbles from each side, we know the scales will still balance.
    ::我们可以对真实物体和对等方程一样做同样的事情。正如我们从等式的两侧减去2个标志一样,我们可以从两侧清除两颗大理石。因为我们从两侧都删除了同样数量的大理石,我们知道这些天平仍然会保持平衡。

    Then, because there are three bags of marbles on the left-hand side of the scale, we can divide the marbles on the right-hand side into three equal piles. You can see that there are nine marbles in each.
    ::然后,由于天平左侧有三袋大理石,我们可以将右手边的大理石分成三堆。你可以看到每块有九块大理石。

    Three bags of marbles balances three piles of nine marbles.
    ::三袋大理石 平衡了三堆九块大理石

    So each bag of marbles balances nine marbles, meaning that each bag contains nine marbles.
    ::所以每袋大理石 平衡九大理石 也就是说每袋都含有九大理石

    Solving for Unknown Values 
    ::解决未知值

    1. Solve 6 ( x + 4 ) = 12 .
    ::1. 解决6(x+4)=12。

    This equation has the x buried in " data-term="Parentheses" role="term" tabindex="0"> parentheses . To dig it out, we can proceed in one of two ways: we can either distribute the six on the left, or divide both sides by six to remove it from the left. Since the right-hand side of the equation is a multiple of six, it makes sense to divide. That gives us x + 4 = 2 . Then we can subtract 4 from both sides to get x = − 2 .
    ::此方程式将 x 埋在括号中 。 要挖出它, 我们可以以两种方式之一进行 : 我们可以将左边的 6 分布在左边, 或者将两边除以 6 来从左边除去它。 由于方程式的右侧是 6 的乘数, 分离是有道理的 。 这让我们的 x+ 4 = 2 。 然后我们可以从两边减去 4 来取 x \\\ 2 。

    2. Solve x − 3 5 = 7 .
    ::2. 解决x-35=7。

    It’s always a good idea to get rid of fractions first. Multiplying both sides by 5 gives us x − 3 = 35 , and then we can add 3 to both sides to get x = 38 .
    ::首先除掉分数总是一个好主意。 将两边乘以5会给我们 x-3=35, 然后我们可以在两边增加3 来获得 x=38 。

    Example
    ::示例示例示例示例

    Example 1
    ::例1

    Solve 5 9 ( x + 1 ) = 2 7 .
    ::解决59(x+1)=27。

    First, we’ll cancel the fraction on the left by multiplying by the reciprocal (the multiplicative inverse).
    ::首先,我们将以对等乘法(乘以倍数反差)取消左侧的分数。

    9 5 ⋅ 5 9 ( x + 1 ) = 9 5 ⋅ 2 7 ( x + 1 ) = 18 35

    ::95-1959(x+1)=95-27(x+1)=1835

    Then we subtract 1 from both sides. ( 35 35 is equivalent to 1.)
    ::然后从双方减去1。 (3535等于1)

    x + 1 = 18 35 x + 1 − 1 = 18 35 − 35 35 x = 18 − 35 35 x = − 17 35

    ::x+1=1835x+1-1=1835-3535x=18-3535x=18-3535x______________________________________________________________________________________________________________________________

    These examples are called two-step equations , because we need to perform two separate operations on the equation to isolate the variable .
    ::这些例子被称为两步方程,因为我们需要在方程上进行两个单独的操作,以孤立变量。

    Review 
    ::回顾

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

    1. 6 x − 1.3 = 3.2
      ::6x-1.3=3.2
    2. 4 ( x + 3 ) = 1
      ::4 (x+3) =1
    3. 5 q − 7 = 2 3
      ::5q-7=23
    4. 3 5 x + 5 2 = 2 3
      ::35x+52=23
    5. 0.1 y + 11 = 0
      ::0.1y+11=0
    6. 5 q − 7 12 = 2 3
      ::5q-712=23
    7. 5 ( q − 7 ) 12 = 2 3
      ::5(q-7)12=23
    8. 33 t − 99 = 0
      ::33-99=0
    9. 5 p − 2 = 32
      ::5p-2=32
    10. 10 y + 5 = 10
      ::10y+5=10
    11. 10 ( y + 5 ) = 10
      ::10(y+5)=10

    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 。