3.5 两侧都有变量的线性等式

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  两侧都有变量的线性线性公式

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  Make a Quick Buck
  ::做一个快速插嘴

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  Using equations in a business context doesn’t only happen to professionals in the business world. People across all professions and situations balance personal budgets, track earnings, and make projections. The example below shows how a college student  might use knowledge of in a business context.
  ::在商业背景下使用方程式不仅发生在商业界的专业人员身上。 不同专业和情况的人平衡个人预算、跟踪收入和预测。 下面的例子显示了大学生如何在商业环境中使用知识。

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  Jared is a college student who needs a little money to help pay for school.  He is a talented artist and wants to  sell posters of his art online . He wants to buy a printer for $1,890 that will print his posters.  Jared  estimates that each poster will cost $6.25 between poster paper and ink.  He plans to sell his posters for $15, and h e  would like to determine his break-even point. A break-even point is when your revenue , the money you make, is equal to your costs. Jared wants to determine how many posters he will need to sell to break even.
  ::贾里德是一个大学生,需要一点钱来帮助支付学费。他是一个有才华的艺术家,想在网上出售他的艺术海报。他想买一台打印机,价值1 890美元,印他的海报。贾里德估计,每张海报在海报纸和墨水之间将花费6.25美元。他打算以15美元的价格出售他的海报,他想确定他的平衡点。一个平衡点是,当你的收入,即你赚的钱,与你的费用相等的时候。贾里德想确定他需要卖多少海报才能实现平等。

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  Linear Equations With Variables on Both Sides Visualized
  ::两侧都有变量的线性线性等式可视化

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  Before you can solve Jared’s dilemma, you will need to better understand linear equations with variables on both sides. So far, all the equations had a known value on one side. You  just had to find the value of  $\begin{array}{r}x\end{array}$  which will give us that value. Linear equations with variables on both sides are much harder to solve mentally.
  ::在你解决贾里德的困境之前,你需要更好地了解线性方程和两边变量。 到目前为止,所有方程的一面都有已知值。 你只需要找到x的值,它能给我们这个值。 具有两边变量的线性方程在精神上很难解决。

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  Use the interactive below to determine the value of unknown values in linear equations with variables on both sides.
  ::使用下方的交互作用来确定线性方程中未知值的值,而线性方程中则有两侧的变量。

  

  INTERACTIVE

  Balancing Linear Equations

  

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  • Cancel out the terms on both sides such that all the variables are in one box and all the constants are in the other.
    ::取消双方的条件,使所有变量都放在一个框内,所有常数都放在另一个框内。
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  • To take out terms already in the boxes, covering them with its additive inverse (e.g. move $-x$  on top of $x$  to cancel them out).
    ::取出框内已有的条款,用其添加剂反向覆盖(例如,在 x 上移- x 以取消它们) 。

  
  

  
  

  
  

  
  

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  Discussion Question
  ::讨论问题

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  What steps did you have to do to solve the equation and what order did you do them in?
  ::为了解开这个方程你采取了什么步骤? 你按什么顺序做了?

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  Make a Quick Buck Continued
  ::快速巴克继续

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  To help Jared, begin by looking at a simpler example. Use the interactive below to explore the relationship between cost to help write an equation that will help Jared.
  ::帮助Jared, 首先从一个简单的例子开始。 使用下面的互动来探索成本之间的关系, 来帮助写出一个能帮助Jared的方程式 。

  

  INTERACTIVE

  Poster Sales

  

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  • Use the slider to increase the number of posters that Jared sells.
    ::使用滑板来增加Jared出售的海报数量。
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  • As you do, the revenue and expense bars will change based on the number of sales.
    ::与您一样,收入和开支栏将随销售额的变化而变化。

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  Break-even cost is determined by the formula  $\begin{array}{r}\text{revenue = variable cost + fixed cost.}\end{array}$
  ::收支相抵费用由公式收入=可变费用+固定费用确定。

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  $\begin{array}{r}\bullet \end{array}$  Revenue:  $\begin{array}{r}15x\end{array}$
  ::收入:15x

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  $\begin{array}{r}\bullet \end{array}$   Variable Cost:  $\begin{array}{r}6.25x\end{array}$
  ::可变费用:6.25x

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  $\begin{array}{r}\bullet \end{array}$  Fixed Cost: 1890
  ::* 固定费用:1890年

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  When you   substitute these into the formula , you  get the following equation:
  ::当您将这些替换为公式时,您可以得到以下方程:

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  $$\begin{array}{r}15x=6.25x+1890\end{array}$$

  
  ::15x=6.25x+1890

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  Up until now, you have only seen equations with variables on one side. To solve an equation with variables on both sides you will need to move all terms with variables to one side of the equation. To do this you will use the same approach that you used to move a constant from one side of the equation to the other: you will add the term’s additive inverse to both sides of the equation.
  ::直到现在,你只看到一边有变量的方程式。要解决一边有变量的方程式。要解决两边有变量的方程式,你需要将所有带有变量的术语移到方程式的一边。要做到这一点,你将使用与从方程式的一边向另一边移动一个常数相同的方法:你将把该词的添加剂反向添加到方程式的两侧。

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  Example
  ::示例示例示例示例

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  $\begin{array}{r}x+2=3x-8\end{array}$
  ::x+2=3x-8

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  In this example, you need to combine the  $\begin{array}{r}x\end{array}$  and the $\begin{array}{r}3x\end{array}$ .  You can either move the $\begin{array}{r}x\end{array}$  to the left or move the $\begin{array}{r}3x\end{array}$  to the right. Both strategies are correct but keeping the variable positive may help prevent silly mistakes. Move  the $\begin{array}{r}x\end{array}$  to the right side of the equation. The additive inverse of  $\begin{array}{r}x\end{array}$   is  $\begin{array}{r}-x\end{array}$  so you will add this to both sides of the equation. The additive inverses are usually lined up vertically with the like terms to make the addition easier visually.
  ::在此示例中, 您需要将 x 和 3x 组合在一起。 您可以将 x 向左移动, 或者将 3x 向右移动。 两种策略都是正确的, 但保持变量正数可能有助于防止愚蠢的错误。 将 x 移动到方程的右侧。 x 的添加反方向是 - x , 这样您就可以将这个添加到方程的两侧。 添加反方向通常会垂直排列, 并用相似的条件使添加更加容易视觉化 。

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  $$\begin{array}{rl}x+2& =3x-8\\ 2& =2x-8\\ 10& =2x\\ 5& =x\end{array}$$

  
  ::x+2=3x-82=2x-810=2x5=x

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  R eturn to the example from the section introduction and solve the equation $\begin{array}{r}15x=6.25x+1890.\end{array}$  F irst, you will need to get the variables on the same side. Begin by s ubtracting  $\begin{array}{r}6.25x\end{array}$   from both sides. From there,  you should be able to solve the resulting equation.
  ::从部分导言返回到示例, 并解析等式 15x=6. 25x+1890。 首先, 您需要在同一侧获取变量。 从两侧减去 6. 25x 开始。 从那里, 您应该能够解析结果的等式 。

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  $$\begin{array}{rl}15x& =6.25x+1890\\ 8.75x& =1890\\ x& =216\end{array}$$

  
  ::15x=6.25x+18908.75x=1890x=216

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  Jared will need to sell 216 posters to break even.
  ::贾里德需要卖掉216张海报才能平分

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  Discussion Questions
  ::讨论问题 讨论问题

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  1. Does this answer make sense within the context? why or why not?
     ::这个答案在上下文中是否有意义?为什么或为什么没有?
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  2. How do you check to make sure that 216 is the correct answer? Is it the correct answer?
     ::你如何确保216是正确的答案?
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  3. How much revenue will Jared earn if he sells 216 posters? What will his profit be?
     ::贾里德卖216张海报能挣多少收入?
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  4. Do you think Jared should raise the sales price, lower the sales price, or that he won’t be able to make a profit at any sales price?
     ::你认为Jared应该提高销售价格, 降低销售价格, 还是他不能以任何销售价格盈利?

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  Perimeter
  ::周边

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       The general steps for solving a multi-step equation with linear equations:
  ::用线性方程解决多步方程的一般步骤:

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  1. Parenthesis - Use the distribut i ve   property to simplify expressions with parenthesis if there are any.
     ::括号 - 使用分配属性简化括号中的表达式(如果有的话)。
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  2. Like Terms - Move all terms with variables to one side of the equation and combine like terms.
     ::类似条件 - 将带有变量的所有条件移动到方程式的一边, 并合并类似条件 。
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  3. 2-Step Equation - Solve the resulting 2-step equation.
     ::2 - 级方程 - 解决由此产生的两步方程。

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  These steps can vary based on the equation, and some strategies will be more efficient in certain situations than in others. When there are situations where other strategies will be more efficient they will be pointed out.
  ::这些步骤可因等式而异,某些战略在某些情况下比在其他情况下更为有效,在出现其他战略效率更高的情况时,将指出这些战略。

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  Multi-step equations are used commonly in situations that involve geometric formulas.
  ::在涉及几何公式的情况下,通常使用多步方程。

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  Example
  ::示例示例示例示例

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  A rectangular parking lot has a width of 51 feet and a perimeter that is five times the length. What is the length of the parking lot?
  ::长方形停车场的宽度为51英尺,周边宽度为5倍。停车场的长度是多少?

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  You can  solve this using the following equation:
  ::您可以使用以下方程式解析 :

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  $$\begin{array}{r}P=2(W+L)\end{array}$$

  
  ::P=2(W+L)

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  $\begin{array}{rl}& \bullet W=51\\ & \bullet L=x\\ & \bullet P=5x\end{array}$  
  ::W=51 l=x l=5x W=51 lL=x P=5x

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  Now, substitute the values into the equation and solve for  $\begin{array}{r}x:\end{array}$  
  ::现在, 将值替换为方程, 并解析 x :

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  $$\begin{array}{rl}5x& =2(51+x)\\ 5x& =102+2x\\ 3x& =102\\ x& =34\end{array}$$

  
  ::5x=2(251+x)5x=102+2x3x=102x=34

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  Answer = 34 feet
  ::答复=34英尺

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  Parenthesis and Like Terms
  ::括号及类似术语

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  L ook at an example that combines everything about linear equations that you have learned so far.
  ::举个例子 综合了所有关于线性方程式的事物 你迄今所学的线性方程式

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  Example
  ::示例示例示例示例

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  $$\begin{array}{rl}3(x+5)& =2(\text{-}5+x)-4x\\ 3x+15& =2(\text{-}5+x)-4x\\ 3x+15& =\text{-}10+2x-4x\\ 3x+15& =\text{-}10-2x\\ 5x+15& =\text{-}10\\ 5x& =\text{-}25\\ x& =\text{-}5\end{array}$$

  
  ::3(xx+5)=2(-5+x)-4x3x+15=2(-5+x)-4x3x+15=-10+2x-4x3xx+15=-10-2x-4x3x+15=-2x5x+15=-2x5x+15=-105x=-25x=-5)

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     Summary
  ::摘要

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  When solving for a variable, remember that an operation on one side of the equation must be performed on the other, to keep the equation balanced.
  ::当解答变量时, 请记住, 必须在方程的一边执行操作, 以使方程保持平衡 。

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  The general steps for solving a multi-step equation:
  ::解决多步方程问题的一般步骤:

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  • Use the distributive property to simplify equations with parenthes e s. 
    ::使用分配财产来简化带有括号的方程。
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  • Combine like terms to simplify expressions.
    ::将类似术语合并以简化表达式 。
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  • Solve for the variable by using inverse operations. 
    ::使用反向操作解决变量 。