4.8 使用斜坡拦截表的图形

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• 选择节使用 斜坡- 截取窗体的图形图

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  使用 斜坡- 截取窗体的图形图

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  Graphs using Slope-Intercept Form 
  ::使用 斜坡- 截取窗体的图形图

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  The total profit of a business is described by the equation $\begin{array}{r}y=15000x-80000,\end{array}$  where $\begin{array}{r}x\end{array}$ is the number of months the business has been running. How much profit is the business making per month, and what were its start-up costs? How much profit will it have made in a year?
  ::企业的总利润由y=15000x-80000等式来描述,其中x是企业经营月数。每月企业的利润是多少,启动成本是多少?一年中它能赚到多少利润?

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  Identifying Slope and  $\begin{array}{r}\mathbf{y}\mathbf{\text{-intercept}}\end{array}$
  
  ::识别斜坡和 Y 界面

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  So far, we’ve been writing a lot of our equations in slope-intercept form - that is, we’ve been writing them in the form $\begin{array}{r}y=mx+b\end{array}$ , where $\begin{array}{r}m\end{array}$ and $\begin{array}{r}b\end{array}$ are both constants. It just so happens that $\begin{array}{r}m\end{array}$ is the and the point $\begin{array}{r}(0,b)\end{array}$ is the $\begin{array}{r}y\text{-intercept}\end{array}$  of the graph of the equation, which gives us enough information to draw the graph quickly.
  ::到目前为止,我们一直在以斜坡界面的形式写下许多方程式,也就是说,我们一直在以y=mx+b的形式写出这些方程式,而m和b是两者的常数。正巧,m是方程式图的y界面,而点(0,b)就是该方程式的y界面,这为我们提供了能够快速绘制图形的足够信息。

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  Identify the slope and $\begin{array}{r}y\text{-intercept}\end{array}$  of the following equations.
  ::确定以下方程的斜坡和Y的中间点。

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  a)  $\begin{array}{r}y=3x+2\end{array}$
  ::a) y=3x+2

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  Comparing $\begin{array}{r}y=3x+2\end{array}$  with  $\begin{array}{r}y=mx+b,\end{array}$  we can see that $\begin{array}{r}m=3\end{array}$ and $\begin{array}{r}b=2\end{array}$ . So $\begin{array}{r}y=3x+2\end{array}$ has a slope of 3 and a $\begin{array}{r}\mathbf{y}\mathbf{\text{-intercept}}\end{array}$ of (0, 2).
  ::将 y= 3x+2 与 y= mx+b 相比,我们可以看到 m= 3 和 b=2 。 所以 y= 3x+2 的斜坡为 3 和 y interception( 0, 2) 。

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  b)  $\begin{array}{r}y=0.5x-3\end{array}$
  :b)y=0.5x-3

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  Again, by comparing this equation with the slope-intercept form, $\begin{array}{r}y=mx+b,\end{array}$  you can see that $\begin{array}{r}y=0.5x-3\end{array}$  has a slope of 0.5 and a $\begin{array}{r}\mathbf{y}\mathbf{\text{-intercept}}\end{array}$   of (0, -3).
  ::同样,通过将这个方程与斜坡截面(y=mx+b)进行比较,你可以看到y=0.5x-3的斜度为0.5,Y-截面为(0,-3)。

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  Notice that the intercept is negative . The $\begin{array}{r}b-\end{array}$ term includes the sign of the operator (plus or minus) in front of the number - for example, $\begin{array}{r}y=0.5x-3\end{array}$ is identical to $\begin{array}{r}y=0.5x+(\text{-}3)\end{array}$ , and that means that $\begin{array}{r}b\end{array}$ is -3, not just 3.
  ::注意拦截是负的。 b- 条件包括操作员在数字前面的标记( 上下) -- 例如, y= 0. 5x-3 与y= 0. 5x+( 3) 相同, 这意味着 b is - 3, 而不仅仅是 3。

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  c)  $\begin{array}{r}y=\text{-}7x\end{array}$
  ::c)y=7x

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  At first glance, this equation doesn’t look like it’s in slope-intercept form , b ut  you can rewrite it as $\begin{array}{r}y=\text{-}7x+0\end{array}$ , and that means it has a slope of -7 and a $\begin{array}{r}\mathbf{y}\mathbf{\text{-intercept}}\end{array}$ of (0, 0). Notice that the slope is negative and the line passes through the origin.
  ::乍一看,这个方程式看起来不像是斜坡界面, 但你可以重写为y= 7x+0, 这意味着它的斜坡为 - 7 和 y interception( 0, 0 ) 。 注意斜坡是负的, 线会穿过原点 。

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  d)  $\begin{array}{r}y=\text{-}4\end{array}$
  ::d)y=4

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  You  can rewrite this one as $\begin{array}{r}y=0x-4,\end{array}$  giving a slope of 0 and a $\begin{array}{r}\mathbf{y}\mathbf{\text{-intercept}}\end{array}$   of (0, -4). This is a horizontal line.
  ::您可以重写此行为 y= 0x-4, 给出0 的斜度和 y interview 值 ( 0, - 4) 。 这是一条水平线 。

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  Identifying Slope and y-intercept  on a Graph 
  ::识别图上的斜坡和 Y 界面

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  Identify the slope and $\begin{array}{r}y\text{-}\end{array}$ intercept of the lines on the graph shown below.
  ::标明下图所显示的线条的斜坡和 Y 交叉点。

  

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  The intercepts have been marked, as well as some convenient lattice points that the lines pass through.
  ::拦截已经标记, 以及一些方便的绳子 点出线经过。

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  a) The $\begin{array}{r}\mathbf{y}\mathbf{\text{-intercept}}\end{array}$  is (0, 5). The line also passes through (2, 3), so the slope is $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{-2}{2}=-1.\end{array}$
  :a) Y 界面是 (0, 5) , 线也通过 (2, 3) , 因此斜坡是 x 22 1 。

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  b) The $\begin{array}{r}\mathbf{y}\mathbf{\text{-intercept}}\end{array}$  is (0, 2). The line also passes through (1, 5), so the slope is $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{3}{1}=3.\end{array}$
  :b) Y 界面是 (0, 2) , 线也通过(1, 5) , 所以斜坡是 x= 31= 3 。

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  c) The $\begin{array}{r}\mathbf{y}\mathbf{\text{-intercept}}\end{array}$  is (0, -1). The line also passes through (2, 3), so the slope is $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{4}{2}=2.\end{array}$
  :c) Y 界面是 (0, - 1) 。线也通过(2, 3),所以斜坡是 x=42=2。

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  d) The $\begin{array}{r}\mathbf{y}\mathbf{\text{-intercept}}\end{array}$  is (0, -3). The line also passes through (4, -4), so the slope is $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{\text{-}1}{4}=\text{-}\frac{1}{4}\end{array}$ or -0.25.
  :d) y-interview is (0, - 3) y-interview 。该线也通过(4, - 4),所以斜坡是 yx= 14= 14 或-0.25。

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  Graphing an Equation in Slope-Intercept Form
  ::以斜坡- 截取窗体绘制方位图

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  Once we know the slope and intercept of a line, it’s easy to graph it. Just remember what slope means. Let's look back at this example from Lesson 4.1.
  ::一旦我们知道一条线的斜坡和截断, 就可以很容易地用图解。 只要记住斜坡的含义。 让我们回顾一下从第4.1课中得出的这个例子 。

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  Ali is trying to work out a trick that his friend showed him. His friend started by asking him to think of a number, then double it, then add five to the result. Ali has written down a rule to describe the first part of the trick. He is using the letter $\begin{array}{r}x\end{array}$ to stand for the number he thought of and the letter $\begin{array}{r}y\end{array}$ to represent the final result of applying the rule. He wrote his rule in the form of an equation: $\begin{array}{r}y=2x+5.\end{array}$
  ::Ali试图找出他朋友给他的诡计。他的朋友首先要求他想一个数字,然后翻一番,然后在结果中增加五个。Ali写了一条规则来描述这个诡计的第一部分。他用字母x来表示他所想的数字,用字母y来表示应用规则的最后结果。他以公式的形式写了他的规则:y=2x+5。

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  Help him visualize what is going on by graphing the function that this rule describes.
  ::帮助他通过绘制此规则所描述的函数的图形 来想象正在发生的事情。

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  In that example, we constructed a table of values, and used that table to plot some points to create our graph.
  ::在这个例子中,我们构建了一个数值表, 并用该表绘制了一些点来创建我们的图表。

  

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  We also saw another way to graph this equation. Just by looking at the equation, we could see that the $\begin{array}{r}y\text{-intercept}\end{array}$  was (0, 5), so we could start by plotting that point. Then we could also see that the slope was 2, so we could find another point on the graph by going over 1 unit and up 2 units. The graph would then be the line between those two points.
  ::我们还看到了用另外一种方法来绘制这个方程。通过查看方程,我们可以看到y-截取是(0, 5),这样我们就可以从绘制那个点开始。然后我们也可以看到斜坡是2,这样我们就可以在图形上找到另一个点,即超过1个单位和上升2个单位。然后,图表将是这两个点之间的直线。

  

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  Here’s another problem where we can use the same method.
  ::还有一个问题, 我们可以使用同样的方法。

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  Graphing a Function
  ::构造函数

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  Graph the following function: $\begin{array}{r}y=\text{-}3x+5\end{array}$
  ::如下函数图示: y=-3x+5

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  To graph the function without making a table, follow these steps:
  ::要在不绘制表格的情况下绘制函数图,请遵循这些步骤:

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  1. Identify the $\begin{array}{r}y\text{-intercept}\end{array}$ : $\begin{array}{r}b=5\end{array}$
     ::识别 Y 界面: b= 5
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  2. Plot the intercept: (0, 5)
     ::截取的绘图: (0, 5)
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  3. Identify the slope: $\begin{array}{r}m=\text{-}3.\end{array}$  (This is equal to $\begin{array}{r}\frac{\text{-}3}{1}\end{array}$ , so the rise is -3 and the run is 1.)
     ::标记斜坡: m=-3。 (这等于 - 31, 上升为 - 3, 运行为 1 )
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  4. Move over 1 unit and down 3 units to find another point on the line: (1, 2).
     ::移动 1 个单位以上, 向下移动 3 个单位, 以在行中找到另一个点 : (1, 2) 。
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  5. Draw the line through the points (0, 5) and (1, 2).
     ::通过点(0、5)和点(1、2)绘制线条。

  

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  Notice that to graph this equation based on its slope, we had to find the rise and run - and it was easiest to do that when the slope was expressed as a fraction . That’s true in general: to graph a line with a particular slope, it’s easiest to first express the slope as a fraction in simplest form, and then read off the numerator and the denominator of the fraction to get the rise and run of the graph.
  ::请注意,要根据斜坡绘制这个方程图,我们必须找到上升并运行 — — 当斜坡以分数表示时,我们最容易这样做。 一般来说,这是真的:用特定斜坡绘制一条线,首先以最简单的形式将斜坡以分数表示,然后从点数和分数分母中读出以获得图形的上升和运行。

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  Finding the Rise and the Run 
  ::寻找升起和奔跑

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  Find integer values for the rise and run of the following slopes, then graph lines with corresponding slopes.
  ::查找以下坡度的上升和运行的整数,然后用相应的斜度绘制图形线。

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  a)  $\begin{array}{r}m=3\end{array}$
  ::a) m=3

  

  

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  b)  $\begin{array}{r}m=-2\end{array}$
  :b) m%%2

  

  

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  Changing the Slope or Intercept of a Line
  ::更改线条的斜曲或截取

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  The following graph shows a number of lines with different slopes, but all with the same $\begin{array}{r}y\text{-intercept}:\end{array}$  (0, 3).
  ::下图显示一些带有不同斜坡的线条,但都有相同的y-interfict: (0, 3) 。

  

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  You can see that all the functions with positive slopes increase as we move from left to right, while all functions with negative slopes decrease as we move from left to right. Another thing to notice is that the greater the slope, the steeper the graph.
  ::你可以看到,随着我们从左向右移动,所有带有正斜度的函数都会增加,而所有带有负斜度的函数则会随着我们从左向右移动而减少。另一点值得注意的是,斜度越大,图形越陡。

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  This graph shows a number of lines with the same slope, but different $\begin{array}{r}y\text{-intercepts.}\end{array}$
  ::此图显示一些带有相同斜度的线条, 但有不同的 y 界面 。

  

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  Notice that changing the intercept simply translates (shifts) the graph up or down. Take a point on the graph of $\begin{array}{r}y=2x\end{array}$ , such as (1, 2). The corresponding point on $\begin{array}{r}y=2x+3\end{array}$ would be (1, 5). Adding 3 to the $\begin{array}{r}y\text{-intercept}\end{array}$  means we also add 3 to every other $\begin{array}{r}y\end{array}$  value on the graph. Similarly, the corresponding point on the $\begin{array}{r}y=2x-3\end{array}$ line would be (1, -1); we would subtract 3 from the $\begin{array}{r}y\end{array}$  value and from every other $\begin{array}{r}y\end{array}$  value.
  ::请注意, 截取器只需向上或向下翻转图形( 变换) 。 在 y= 2x 图形上取一个点, 例如(1, 2) 。 在 y= 2x+3 上的相应点将是(1, 5) 。 在 y 拦截中添加 3 意味着我们也将在图上的其他值中每增加 3 y 值。 同样, y= 2x-3 行上的相应点将是(1, -1) ; 我们将从 y 值和 y 值中减去 3 。

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  Notice also that these lines all appear to be parallel. Are they truly parallel?
  ::也注意到这些线条看起来都是平行的。 它们真的平行吗 ?

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  To answer that question, we’ll use a technique that you’ll learn more about in a later chapter. We’ll take 2 of the equations - say, $\begin{array}{r}y=2x\end{array}$ and $\begin{array}{r}y=2x+3\end{array}$  - and solve for values of $\begin{array}{r}x\end{array}$ and $\begin{array}{r}y\end{array}$ that satisfy both equations. That will tell us at what point those two lines intersect, if any. (Remember that parallel lines , by definition, are lines that don’t intersect.)
  ::为了回答这个问题,我们将使用一种你将在后一章中学到更多知识的方法。 我们将使用两个方程式中的两个方程式 — — 比如 y=2x 和 y=2x+3 — — 并解决满足两个方程式的 x 和 y 的值。 这将告诉我们这两条线在哪个点交叉(如果有的话 ) 。 (记住,根据定义,平行线条是不会交叉的线条 。 )

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  So what values would satisfy both $\begin{array}{r}y=2x\end{array}$ and $\begin{array}{r}y=2x+3?\end{array}$  Well, if both of those equations were true, then $\begin{array}{r}y\end{array}$ would be equal to both $\begin{array}{r}2x\end{array}$ and $\begin{array}{r}2x+3,\end{array}$  which means those two expressions would also be equal to each other. So we can get our answer by solving the equation $\begin{array}{r}2x=2x+3.\end{array}$
  ::那么什么值能满足y=2x 和y=2x+3 。 那么, 如果这两个方程都是真实的, 那么y将等于 2x 和 2x+3 , 这意味着这两个表达式也等于 彼此。 这样我们就能通过解答 2x=2x+3 的方程获得答案 。

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  W hat happens when we  try to solve that equation? If we subtract $\begin{array}{r}2x\end{array}$ from both sides, we end up with $\begin{array}{r}0=3\end{array}$ . That can’t be true no matter what $\begin{array}{r}x\end{array}$ equals. And that means that there just isn’t any value for $\begin{array}{r}x\end{array}$ that will make both of the equations we started out with true. In other words, there isn’t any point where those two lines intersect. They are parallel, just as we thought.
  ::当我们试图解决这个等式时会怎样?如果我们从两边减去2x,我们最终会以 0=3 来结束。 这不可能是真实的,不管x等于什么。 这意味着 x 没有任何价值能让我们开始的两个等式都变成真实的。 换句话说,没有两条线相交点。 它们是平行的,就像我们想象的那样。

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  W e’d find out the same thing no matter which two lines we’d chosen. In general, since changing the intercept of a line just results in shifting the graph up or down, the new line will always be parallel to the old line as long as the slope stays the same.
  ::不管我们选择了哪条线,我们都会发现同样的事情。 一般来说,由于改变一条线的拦截方式只会导致图的上下移动,只要斜坡保持不变,新线将永远与旧线平行。

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  Any two lines with identical slopes are parallel.
  ::具有相同斜坡的任何两条线都是平行的。

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  Examples
  ::实例

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  Find integer values for the rise and run of the following slopes, then graph lines with corresponding slopes.
  ::查找以下坡度的上升和运行的整数,然后用相应的斜度绘制图形线。

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  Example 1
  ::例1

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  $\begin{array}{r}m=0.75\end{array}$
  ::m=0.75

  

  

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  Example 2
  ::例2

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  $\begin{array}{r}m=-.375\end{array}$
  ::3075 má. 375

   

  

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  Review 
  ::回顾

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  Identify the slope and $\begin{array}{r}y\text{-intercept}\end{array}$  for the following equations.
  ::标明以下方程的斜坡和 Y 界面。

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  1. $\begin{array}{r}y=2x+5\end{array}$
     ::y=2x+5 y=2x+5
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  2. $\begin{array}{r}y=\text{-}0.2x+7\end{array}$
     ::y=-0.2x+7 y=-0.2x+7
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  3. $\begin{array}{r}y=x\end{array}$
     ::y=x y=x
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  4. $\begin{array}{r}y=3.75\end{array}$
     ::y=3.75 y=3.75

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  Identify the slope of the following lines.
  ::标明下列线条的斜坡。

  5. 

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  Identify the slope and $\begin{array}{r}y\text{-intercept}\end{array}$  for the following functions.
  ::标明下列函数的斜坡和 Y 界面。

  6. 

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  For 7-10, plot the following functions on a graph.
  ::7-10时,在图表中绘制下列函数。

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  7. $\begin{array}{r}y=2x+5\end{array}$
     ::y=2x+5 y=2x+5
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  8. $\begin{array}{r}y=\text{-}0.2x+7\end{array}$
     ::y=-0.2x+7 y=-0.2x+7
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  9. $\begin{array}{r}y=x\end{array}$
     ::y=x y=x
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  10. $\begin{array}{r}y=3.75\end{array}$
      ::y=3.75 y=3.75

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  11. Which two of the following lines are parallel?
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      1. $\begin{array}{r}y=2x+5\end{array}$
         ::y=2x+5 y=2x+5
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      2. $\begin{array}{r}y=\text{-}0.2x+7\end{array}$
         ::y=-0.2x+7 y=-0.2x+7
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      3. $\begin{array}{r}y=x\end{array}$
         ::y=x y=x
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      4. $\begin{array}{r}y=3.75\end{array}$
         ::y=3.75 y=3.75
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      5. $\begin{array}{r}y=\text{-}\frac{1}{5}x-11\end{array}$
         ::y=-15x-11 y=-15x-11
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      6. $\begin{array}{r}y=\text{-}5x+5\end{array}$
         ::y=-5x+5 y=-5x+5
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      7. $\begin{array}{r}y=\text{-}3x+11\end{array}$
         ::y=-3x+11 y=-3x+11
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      8. $\begin{array}{r}y=3x+3.5\end{array}$
         ::y=3x+3.5 y=3x+3.5
      
      ::以下两行的哪个线平行? y=2x+5y=-0.2x+7y=xy=3. 75y=-15x-11y=-5x+5y=-3x+11y=3x+3.5
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  12. What is the $\begin{array}{r}y\text{-intercept}\end{array}$  of the line passing through (1, -4) and (3, 2)?
      ::横穿线(1,4)和(3,2)的 Y 截断点是什么?
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  13. What is the $\begin{array}{r}y\text{-intercept}\end{array}$  of the line with slope -2 that passes through (3, 1)?
      ::斜坡 -2 的横线( 3, 1) 的 Y 插点是什么 ?
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  14. Line $\begin{array}{r}A\end{array}$ passes through the points (2, 6) and (-4, 3). Line $\begin{array}{r}B\end{array}$ passes through the point (3, 2.5), and is parallel to line $\begin{array}{r}A\end{array}$
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      1. Write an equation for line $\begin{array}{r}A\end{array}$ in slope-intercept form.
         ::以斜坡界面的形式为 A 线写一个方程式 。
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      2. Write an equation for line $\begin{array}{r}B\end{array}$ in slope-intercept form.
         ::以斜坡界面的形式为 B 线写一个方程式 。
      
      ::A线穿过点(2、6和4、3),B线穿过点(3、2.5),与A线平行,以斜度截取形式写出A线的方程式。B线写出斜度截取形式的方程式。
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  15. Line $\begin{array}{r}C\end{array}$ passes through the points (2, 5) and (1, 3.5). Line $\begin{array}{r}D\end{array}$ is parallel to line $\begin{array}{r}C\end{array}$ , and passes through the point (2, 6). Name another point on line $\begin{array}{r}D\end{array}$ . (Hint: you can do this without graphing or finding an equation for either line.)
      ::线C通过点(2,5)和点(1,3.5)。 线D与线C平行,通过点(2,6),在D线上另请一个点(提示:您可以这样做,而不必为两行中的两行绘制图表或找到方程式)。

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  Review (Answers)
  ::回顾(答复)

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