2.4 找出平行线的等同

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  寻找平行线的平方

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  Pablo is researching different types of exercise to see which gives the best calorie burn. He found that an indoor cycling class can burn up to 150 calories every 10 minutes. The elliptical machine burns 270 calories every 30 minutes. Do these two forms of exercise burn the same amount of calories in an hour?
  ::Pablo正在研究不同种类的锻炼活动,以了解哪种锻炼能产生最佳的热量燃烧。 他发现室内自行车类可以每10分钟燃烧150卡路里。 椭圆机每30分钟燃烧270卡路里。 这两种锻炼形式在一小时内燃烧同样的热量吗?

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  Equation of Parallel Lines
  ::平行线的等式

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  When two lines are parallel, they have the same and never intersect. So, if a given line has a slope of -2, then any line that is parallel to that line will also have a slope of -2, but it will have a different $\begin{array}{r}y-\end{array}$ intercept .
  ::当两条线平行时,两条线是相同的,从不交叉。因此,如果给定线的斜度为 -2,那么与该线平行的任何线的斜度也将是 -2,但它会有不同的y-interinter。

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  Let's find the equation of the following parallel lines.
  ::让我们找到以下平行线的等式。

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  1. Find the equation of the line that is parallel to $\begin{array}{r}y=\frac{2}{3}x-5\end{array}$ and passes through (-12, 1).
     ::查找与 y= 23x- 5 平行的直线的方程式,然后通过 (12, 1) 。

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  We know that the slopes will be the same; however we need to find the $\begin{array}{r}y-\end{array}$ intercept for this new line. Use the point you were given, (-12, 1) and plug it in for $\begin{array}{r}x\end{array}$ and $\begin{array}{r}y\end{array}$ to solve for $\begin{array}{r}b\end{array}$ .
  ::我们知道斜坡会是一样的; 但是我们需要找到新线的 y- interview 。 使用您给定的点, (12, 1) 并插入 x 和 y 来解析 b 。

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  $$\begin{array}{rl}y& =\frac{2}{3}x+b\\ 1& =\frac{2}{3}(-12)+b\\ 1& =-8+b\\ 9& =b\end{array}$$

  
  ::y=23x+b1=23(- 12)+b1=8+b9=b

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  The equation of the parallel line is $\begin{array}{r}y=\frac{2}{3}x+9\end{array}$ .
  ::平行线的方程式是y=23x+9。

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  2. Find the equation of the line that passes through (4, -7) and is parallel to $\begin{array}{r}y=-2\end{array}$ .
     ::查找通过线( 4, - 7) 并与 y 2 平行的线的方程。

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  The line $\begin{array}{r}y=-2\end{array}$ does not have a $\begin{array}{r}x-\end{array}$ term , meaning it has no slope. This is a horizontal line. Therefore , to find the horizontal line that passes through (4, -7), we only need the $\begin{array}{r}y-\end{array}$ coordinate. The line would be $\begin{array}{r}y=-7\end{array}$ .
  ::y% 2 的线没有 x - term, 意思是它没有斜度。 这是一条水平线。 因此, 要找到通过( 4, - 7) 的水平线, 我们只需要 Y - 坐标。 这条线将是 y = 7 。

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  The same would be true for vertical lines, but all vertical line equations are in the form $\begin{array}{r}x=a\end{array}$ . The $\begin{array}{r}x-\end{array}$ coordinate of a given point would be what is needed to determine the equation of the parallel vertical line.
  ::垂直线也是一样,但所有垂直线方程式都是表x=a。 确定平行垂直线方程式所需要的是某一点的x-坐标。

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  3. Find the equation of the line that passes through (6, -10) and is parallel to the line that passes through (4, -6) and (3, -4).
     ::查找通过线(6,10)的方程式,该方程式与通过线(4,6)和(3,4)的方程式平行。

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  First, we need to find the slope of the line that our line will be parallel to. Use the points (4, -6) and (3, -4) to find the slope.
  ::首先,我们需要找到一条线的斜坡, 这条线将和这条线平行。 使用点( 4, 6) 和点( 3, 4) 来找到斜坡 。

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  $$\begin{array}{r}m=\frac{-4-(-6)}{3-4}=\frac{2}{-1}=-2\end{array}$$

  
  ::m*4 - (-6)3 - 4=2 - 1 *2

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  This is the slope of our given line as well as the parallel line. Use the point (6, -10) to find the $\begin{array}{r}y-\end{array}$ intercept of the line that we are trying to find the equation for.
  ::这是我们给定线的斜坡和平行线。 使用点( 6, - 10) 来找到我们试图找到方程的 y- 中间线 。

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  $$\begin{array}{rl}-10& =-2(6)+b\\ -10& =-12+b\\ 2& =b\end{array}$$

  
  ::- 102(6)+b-1012+b2=b

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  The equation of the line is $\begin{array}{r}y=-2x+2\end{array}$ .
  ::线条的方程式是 y2x+2 。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked if the two forms of exercise burn the same amount of calories in an hour. 
  ::早些时候,有人问您,两种形式的锻炼是否在一小时内燃烧同样数量的卡路里。

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  The equation for indoor cycling is $\begin{array}{r}y=\frac{150}{10}x\end{array}$ or $\begin{array}{r}y=15x\end{array}$ . For the elliptical, the equation would be $\begin{array}{r}y=\frac{270}{30}x\end{array}$ or $\begin{array}{r}y=9x\end{array}$ . The two lines are not parallel, so the two exercises do not burn an equal number of calories. After 60 minutes, indoor cycling will burn 900 calories and the elliptical will burn 540 calories.
  ::室内自行车的方程是y=15010x或y=15x。对椭圆而言,方程是y=27030x或y=9x。两条线不平行,因此两条练习不会燃烧同等数量的卡路里。60分钟后,室内自行车将燃烧900卡路里,而椭圆将燃烧540卡路里。

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  Example 2
  ::例2

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  Find the equation of the line that is parallel to $\begin{array}{r}x-2y=8\end{array}$ and passes through (4, -3).
  ::查找与 x-2y=8 平行的直线的方程式,然后通过(4, 3) 。

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  First, we need to change this line from standard form to slope-intercept form .
  ::首先,我们需要将这条线从标准格式改为斜坡拦截形式。

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  $$\begin{array}{rl}x-2y& =8\\ -2y& =-x+8\text{Now, we know the slope is}\frac{1}{2}.\text{Let's find the new}y-\text{intercept.}\\ y& =\frac{1}{2}x-4\end{array}$$

  
  ::X-2y=8- 2yx+8 现在,我们知道斜坡是12。 让我们找到新的 y- 截取。 y=12x- 4

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  $$\begin{array}{rl}-3& =\frac{1}{2}(4)+b\\ -3& =2+b\\ -5& =b\end{array}$$

  
  ::- 3=12(4)+b-3=2+b-5=b

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  The equation of the parallel line is $\begin{array}{r}y=\frac{1}{2}x-5\end{array}$ or $\begin{array}{r}x-2y=10\end{array}$ .
  ::平行线的方程式是 y= 12x-5 或 x- 2y= 10。

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  Example 3
  ::例3

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  Find the equation of the line that is parallel to $\begin{array}{r}x=9\end{array}$ and passes through (-1, 3).
  ::查找与 x=9 平行的线条的方程,然后通过 (-1, 3) 。

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  $\begin{array}{r}x=9\end{array}$ is a vertical line that passes through the $\begin{array}{r}x-\end{array}$ axis at 9. Therefore, we only need to $\begin{array}{r}x-\end{array}$ coordinate of the point to determine the equation of the parallel vertical line. The parallel line through (-1, 3) would be $\begin{array}{r}x=-1\end{array}$ .
  ::x=9 是一条垂直线,9点时穿过 x - 轴线, 因此, 我们只需要x - 协调点来确定平行垂直线的方程。 通过 (-1, 3) 的平行线将是 x *1 。

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  Example 4
  ::例4

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  Find the equation of the line that passes through (-5, 2) and is parallel to the line that passes through (6, -1) and (1, 3).
  ::寻找通过( 5, 2) 的线的方程, 并平行于通过( 6, 1) 和(1, 3) 的线的方程。

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  First, find the slope between (6, -1) and (1, 3).
  ::首先,找到在(6,-1)和(1,3)之间的斜坡。

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  $$\begin{array}{r}m=\frac{-1-3}{6-1}=\frac{-4}{5}=-\frac{4}{5}\end{array}$$

  
  ::m1-36 -1 4545

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  This will also be the slope of the parallel line. Use this slope with the given point, (-5, 2).
  ::这也将是平行线的斜坡。使用该斜坡时使用给定点( 5, 2) , (5, 5) 。

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  $$\begin{array}{rl}2& =-\frac{4}{5}(-5)+b\\ 2& =4+b\\ -2& =b\end{array}$$

  
  ::245(-5)+b2=4+b-2=b

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  The equation of the parallel line is $\begin{array}{r}y=-\frac{4}{5}x-2.\end{array}$
  ::平行线的方程式是y45x-2。

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  Review
  ::回顾

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  Find the equation of the line, given the following information. You may leave your answer in slope-intercept form.
  ::查找线条的方程式, 给出以下信息。 您可以将答案保留在斜坡界面中 。

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  1. Passes through (4, 7) and is parallel to $\begin{array}{r}x-y=-5\end{array}$ .
     ::通过(4、7),与x-y5平行。
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  2. Passes through (-6, -2) and is parallel to $\begin{array}{r}y=4\end{array}$ .
     ::通过(-6, 2),与y=4平行。
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  3. Passes through (-3, 5) and is parallel to $\begin{array}{r}y=-\frac{1}{3}x-1\end{array}$ .
     ::通过(-3, 5), 与 y13x-1 平行 。
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  4. Passes through (1, -9) and is parallel to $\begin{array}{r}x=8\end{array}$ .
     ::通过(1, - 9),与x=8平行。
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  5. Passes through the $\begin{array}{r}y-\end{array}$ intercept of $\begin{array}{r}2x-3y=6\end{array}$ and parallel to $\begin{array}{r}x-4y=10\end{array}$ .
     ::通过 2x-3y=6 和 x-4y=10 平行的 y - interview 通过 y - interview 。
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  6. Passes through (-12, 4) and is parallel to $\begin{array}{r}y=-3x+5\end{array}$ .
     ::通过( 12, 4) , 与 y3x+5 平行 。
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  7. Passes through the $\begin{array}{r}x-\end{array}$ intercept of $\begin{array}{r}2x-3y=6\end{array}$ and parallel to $\begin{array}{r}x+4y=-3\end{array}$ .
     ::通过x-3y=6的x- interview 且与 x+4y+3 平行的 x- interview 。
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  8. Passes through (7, -8) and is parallel to $\begin{array}{r}2x+5y=14\end{array}$ .
     ::通过(7,-8),与2x+5y=14平行。
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  9. Passes through (1, 3) and is parallel to the line that passes through (-6, 2) and (-4, 6).
     ::通过(1,3),与通过(6,2)和(4,6)的线平行。
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  10. Passes through (-18, -10) and is parallel to the line that passes through (-2, 2) and (-8, 1).
      ::通过(18,-10),与通过(-2,2)和(-8,1)的线平行。
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  11. Passes through (-4, -1) and is parallel to the line that passes through (15, 7) and (-1, -1).
      ::通过(4-4-1),与通过(15、7)和(1-1)的线平行。

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  Are the pairs of lines parallel? Briefly explain how you know.
  ::线条是平行的吗?

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  12. $\begin{array}{r}x-2y=4\end{array}$ and $\begin{array}{r}-5x+10y=16\end{array}$
      ::x-2y=4和-5x+10y=16
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  13. $\begin{array}{r}3x+4y=-8\end{array}$ and $\begin{array}{r}6x+12y=-1\end{array}$
      ::3x+4y8和6x+12y1
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  14. $\begin{array}{r}5x-5y=20\end{array}$ and $\begin{array}{r}x+y=7\end{array}$
      ::5x-5y=20和x+y=7
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  15. $\begin{array}{r}8x-12y=36\end{array}$ and $\begin{array}{r}10x-15y=-15\end{array}$
      ::8-12y=36和10-15y15

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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 。