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  • 平行线线

    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? In this section, we consider equations of lines that have the same or rate of change . 

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    Parallel Lines
    ::平行线线

    Two lines are parallel  if  they never intersect. The two lines will never intersect if their rate of change or slope is the same. For example, 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.
    ::两条线如果从不交叉,则平行。如果两条线的变动率或斜度相同,则它们永远不会交叉。例如,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。

      Parallel Lines
    ::平行线线

    Two lines are parallel if their slopes are equal, that is, if  m 1 = m 2 . 

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    In the image above the lines in each set  would be considered parallel. As we can see each set of parallel lines has the same slope. Parallel  lines can have negative slopes (A in image above), positive slopes (B in image above), slopes of zero (C in image above), or undefined slopes (D in image above).
    ::在每组线条上方的图像中,每组线条将被视为平行的。 我们可以看到每组平行线条都有相同的斜度。 平行线条可能有负斜度( 图像为A) 、 正斜度( 图像为B) 、 零斜度( 图像为C) 或未定义的斜度( 图像为D) 。

    Example 1
    ::例1

    Pablo's research showed that an indoor cycling class can burn up to 150 calories every 10 minutes. The elliptical machine burns 270 calories every 30 minutes. Determine if the lines that model calories burned for these two activities are parallel. Do these two forms of exercise burn the same amount of calories in an hour?
    ::Pablo的研究表明,室内自行车班每10分钟可燃烧150卡路里。 椭圆机每30分钟可燃烧270卡路里。 确定这两种活动所燃烧的模范卡路里线是否平行。 这两种形式的锻炼是否在一小时内燃烧同样的卡路里?

    Solution: The rate for indoor cycling is 150 calories per 10 minutes, so the slope is  150 10 or 15 calories/minute. The equation for indoor cycling is y = 150 10 x    or  y = 15 x . The rate of change is 9 calories per 1 minute. For the elliptical machine, the equation would be y = 270 30 x   or y = 9 x .  Since the or slopes of the lines are not equal, the two lines are not parallel.
    ::溶解度: 室内自行车的速率是每10分钟150卡路里, 坡度是15010或15卡路里/ 分钟。 室内自行车的方程式是y=15010x或y=15x。 变化率是每1分钟9卡路里。 椭圆机的等式是y=27030x或y=9x。 由于线的斜坡不相等, 这两条线不平行 。

    The two exercises do not burn the same number of calories in an hour. After 1 hour or 60 minutes, Pablo will burn 900 calories with indoor cycling and 540 calories with the elliptical machine.
    ::在1小时或60分钟后,巴勃罗将用室内自行车燃烧900卡路里,用椭圆机燃烧540卡路里。

    This video by CK-12 demonstrates how to write and solve linear equations of parallel lines for an application .
    ::CK-12的这段视频展示了如何为应用程序写入和解析平行线条线性方程式的线性方程式。

    Example 2
    ::例2

    Are  y + 5 = − 1 2 ( x + 3 )  and  y = − 1 2 x + 4  parallel?
    ::y+512( x+3) 和 y12x+4 平行吗 ?

    Solution:  To determine if two lines are parallel, we compare their slopes. The slope of the first line is − 1 2  and the slope of the second line is  − 1 2 . These lines have equal slopes, therefore they are parallel. 
    ::解决方案: 要确定两条线是平行的, 我们比较两条线的斜坡。 第一条线的斜坡是 - 12, 第二条线的斜坡是 - 12。 这些线的斜坡是相等的, 因此是平行的 。

    Example 3
    ::例3

    Find the equation of the line that is parallel to y = 2 3 x − 5 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) 。

    Solution: We know that the slopes will be the same and we have a point on the line, so we use point-slope form: 

    y − 1 = 2 3 ( x + 12 )
    If we convert the line to slope-intercept form , we have  y = 2 3 x + 9 .
    ::解答: 我们知道斜坡是相同的, 我们在线上有一个点, 所以我们使用点窗体: y-1=23(x+12) 如果我们将线转换为斜坡界面, 我们就会有 y=23x+9 。

    This video by CK-12 demonstrates how to write equations of parallel lines given a line and one point. 
    ::CK-12的这段视频展示了如何根据一条线和一个点来写出平行线的方程式。

    Example 4
    ::例4

    Find the equation of the line that is parallel to x − 2 y = 8 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) 。

    Solution: First, we need to change this line from general  form to slope-intercept form so we can identify the slope.
    ::解决办法:首先,我们需要将这条线从一般形式改为斜坡拦截形式,这样我们才能识别斜坡。

    x − 2 y = 8 − 2 y = − x + 8 y = 1 2 x − 4

    ::x-2y=8--2yx+8y=12x-4

    Now, we know the slope is 1 2 . We can use point-slope form or we can use slope-intercept form and solve for b . We do the latter below.
    ::现在,我们知道斜坡是12, 我们可以使用点窗体, 或者我们可以使用斜坡截面表, 解决 b. 我们下面做后一个。

    − 3 = 1 2 ( 4 ) + b − 3 = 2 + b − 5 = b

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

    The equation of the parallel line is y = 1 2 x − 5 or in general form  x − 2 y = 10 .
    ::平行线的方程式为y=12x-5,或一般形式为 x-2y=10。

    Example 5
    ::例5

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

    Solution:   x = 9 is a vertical line that passes through the x- axis at 9. Therefore, we only need to use the x- coordinate of the point to determine the equation of the parallel vertical line. The parallel line through (-1, 3) would be x = − 1 .
    ::解析度 : x=9 是一条垂直线, 9点时穿过 x 轴。 因此, 我们只需要使用点的x 坐标来确定平行垂直线的方程。 通过 (-1 、 3) 的平行线将是 x 1 。

    Feature: A Road Map to Healing
    ::特色:实现治疗的路线图

    by Jen Kershaw
    ::由Jen Kershaw著

    Surgery requires many specialized tools and instruments. Parallel lines help in the operating room too.
    ::外科手术需要许多专门工具和工具,平行线也帮助手术室。

    In 1861, Karl Langer, an Austrian anatomist, discovered that by drawing lines in a specific way on a map of the human body, the lines would be parallel to the underlying muscle fibers. Although Langer is credited with this discovery, he acknowledged that French surgeon Baron Dupuytren was the first to notice this. You can see Langer's lines drawn on the image of the head and neck below.
    ::1861年,奥地利解剖学家Karl Langer发现,通过在人体地图上以特定方式绘制线条,这些线条将与底部肌肉纤维平行。 尽管Langer发现了这一发现,但他承认法国外科医生Baron Dupuytren是第一个注意到这一点的。你可以看到Langer的线条是从下面头部和颈部的图像中绘制的。

    Langer's lines gave him a guideline before surgery—a trail to follow as he worked. Today, surgeons, especially those working in cosmetic or forensic surgery, still use Langer's lines as a blueprint to help them understand where the muscle fibers are in the body. This discovery has led to the development of specific surgical techniques and has helped surgeons locate nerves and organs for medical procedures. It has also been found that incisions made parallel to Langer's lines may heal better, although this has been recently debated, as yo u can see in the video below. 
    ::在手术前,Langer的手术线为他提供了指导,这是他工作时要遵循的一条线索。今天,外科医生,特别是那些从事化妆或法医手术的外科医生,仍然以Langer的手术线作为蓝图,帮助他们了解身体肌肉纤维的位置。这一发现导致了具体的外科技术的发展,并帮助外科医生找到用于医疗手术的神经和器官。还发现,与Langer的手术线平行的切口可能愈合得更好,尽管最近已经对此进行过辩论,如下文所示。

    This video by WileyVideoAbstracts discusses the debate regarding Langer's lines.
    ::这段影片由WileyVideoAbspects拍摄,

    Summary
    ::摘要

    • Parallel lines are lines that never intersect.
      ::平行线是永不交叉的线条。
    • We can identify parallel lines by determining if their slopes are equal. 
      ::我们可以通过确定它们的斜坡是否相等来确定平行线。

    Review
    ::回顾

    Are the pairs of lines parallel? 
    ::线条是平行的吗?

    1.  x − 2 y = 4 and − 5 x + 10 y = 16
    ::1. x-2yy=4和-5x+10y=16

    2.  3 x + 4 y = − 8 and 6 x + 12 y = − 1
    ::2. 3x+4y8和6x+12y1

    3.  5 x − 5 y = 20 and x + y = 7
    ::3. 5x-5y=20和x+y=7

    4.  8 x − 12 y = 36 and 10 x − 15 y = − 15
    ::4. 8-12y=36和10-15y15

    Find the equation of the line, given the following information. 
    ::查找线条的方程, 给出以下信息 。

    5. Passes through (-6, -2) and is parallel to y = 4 .
    ::5. 通过(-6, 2),与y=4平行。

    6. Passes through (1, -9) and is parallel to x = 8 .
    ::6. 通过(1,-9),与x=8平行。

    7. Passes through (-3, 5) and is parallel to y = − 1 3 x − 1 .
    ::7. 通过(-3,5),与y13x-1平行。

    8. Passes through (4, 7) and is parallel to x − y = − 5 .
    ::8. 通过(4、7),与x-y5平行。

    9. Passes through (1, 3) and is parallel to the line that passes through (-6, 2) and (-4, 6).
    ::9. 通过(1、3),与通过(6、2)和(4、6)的线平行。

    10. Passes through (-18, -10) and is parallel to the line that passes through (-2, 2) and (-8, 1).
    ::10. 通过(18,-10),与通过(-2,2)和(-8,1)的线平行。

    Explore More
    ::探索更多

    1.  Suppose a coordinate plane were transposed over the map of a city, and Main Street had the equation y = 2 x + 5 . If Broad Street ran parallel to Main Street, and it passed through the point (3, 14), what would be the equation of Broad Street?  
    ::1. 假设一架坐标飞机被移到城市地图上,而主街的等式是y=2x+5,如果布罗德街与主街平行,它经过点(3、14),布罗德街的等式是什么?

    2.  On a model boat, the stack is 8 inches high. The actual stack is 6 feet tall. How tall is the mast on the model if the actual mast is 40 feet tall?
    ::2. 在一艘型船上,船底8英寸高,实际车架6英尺高,如果船顶40英尺高,船顶有多高?

    3.  The amount of money charged for a classified advertisement is directly proportional to the length of the advertisement. If a 50-word advertisement costs $11.50, what is the cost of a 70-word advertisement?
    ::3. 机密广告的收费数额与广告的长度成正比,如果50字广告的费用为11.50美元,那么70字广告的费用是多少?

    4. Are the points (-2,4), (-1,0), (6,5), and (7,2) the vertices of a rectangle? (HINT: Recall a rectangle has pairs of parallel sides.)

    ::4. 要点(-2、4、1、10、6、5和7、2)是矩形的顶点吗? (HINT:回顾矩形有平行的两面。 )

    Answers for Review and Explore More Problems
    ::回顾和探讨更多问题的答复

    Please see the Appendix. 
    ::请参看附录。

    PLIX
    ::PLIX

    Try these interactives that reinforce the concepts explored in this section:
    ::尝试这些强化本节所探讨概念的交互作用 :