3.8 坐标平面上的平行线

章节大纲

• 选择节常规

  折叠 展开

  常规

  全部折叠 展开全部

  • 选择活动新闻通告

    

    新闻通告 讨论区
• 选择节坐标平面的平行线

  折叠 展开

  坐标平面的平行线

  播放段落

  Parallel Lines in the Coordinate Plane
  ::坐标平面的平行线

  播放段落

  Parallel lines are two lines that never intersect. In the coordinate plane , that would look like this:
  ::平行线是两条从不交叉的线条。在坐标平面上,这看起来像:

  

  播放段落

  If we take a closer look at these two lines, the slopes are both $\begin{align*}\frac{2}{3}\end{align*}$ .
  ::如果我们仔细看看这两条线, 斜坡是23。

  播放段落

  This can be generalized to any pair of parallel lines. Parallel lines always have the same slope and different $\begin{align*}y-\end{align*}$ intercepts .
  ::平行线总是有相同的斜坡和不同的y-inter概念。

  播放段落

  What if you were given two parallel lines in the coordinate plane? What could you say about their slopes?
  ::如果在座标飞机上给你两条平行线呢?

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Find the equation of the line that is parallel to $\begin{align*}y=\frac{1}{4}x+3\end{align*}$ and passes through (8, -7).
  ::查找与 y= 14x+3 平行的直线的方程式,然后通过 (8, - 7) 。

  播放段落

  We know that parallel lines have the same slope, so the line will have a slope of $\begin{align*}\frac{1}{4}\end{align*}$ . Now, we need to find the $\begin{align*}y-\end{align*}$ intercept. Plug in 8 for $\begin{align*}x\end{align*}$ and -7 for $\begin{align*}y\end{align*}$ to solve for the new $\begin{align*}y-\end{align*}$ intercept $\begin{align*}(b)\end{align*}$ .
  ::我们知道平行线具有相同的斜坡,所以线的斜坡为14, 现在我们需要找到y- intercept。 x 和 - 7 的插件在 8 中插入, y 中插入, y 中插入, 用于新- intercept (b) 的解答 。

  播放段落

  $$\begin{align*}-7 & = \frac{1}{4}(8)+b\\ -7 & = 2 + b\\ -9 & = b\end{align*}$$
  ::-7=14(8)+b-7=2+b-9=b

  播放段落

  The equation of the parallel line is $\begin{align*}y = \frac{1}{4}x-9\end{align*}$ .
  ::平行线的方程式是 Y= 14x- 9 。

  播放段落

  Example 2
  ::例2

  播放段落

  Are the lines $\begin{align*}3x+4y=7\end{align*}$ and $\begin{align*}y=\frac{3}{4}x+1\end{align*}$ parallel?
  ::行 3x+4y=7 和 Y= 34x+1 平行吗 ?

  播放段落

  First we need to rewrite the first equation in slope-intercept form.
  ::首先我们需要重写第一个方程式 以斜坡界面的形式。

  播放段落

  $$\begin{align*}3x+4y&=7\\4y&=-3x+7\\y&=-\frac{3}{4}x+\frac{7}{4}\end{align*}$$ .
  ::3x+4y=74y3x+7y34x+74。

  播放段落

  The slope of this line is $\begin{align*}-\frac{3}{4}\end{align*}$ while the slope of the other line is $\begin{align*}\frac{3}{4}\end{align*}$ . Because the slopes are different the lines are not parallel.
  ::这条线的斜坡是-34,另一条线的斜坡是34,因为斜坡不同,两条线不平行。

  播放段落

  Example 3
  ::例3

  播放段落

  Find the equation of the line that is parallel to $\begin{align*}y=-\frac{1}{3}x+4\end{align*}$ and passes through (9, -5).
  ::查找与 y13x+4 平行的直线的方程式,然后通过 (9,5) 。

  播放段落

  Recall that the equation of a line is $\begin{align*}y = mx + b\end{align*}$ , where $\begin{align*}m\end{align*}$ is the slope and $\begin{align*}b\end{align*}$ is the $\begin{align*}y-\end{align*}$ intercept. We know that parallel lines have the same slope, so the line will have a slope of $\begin{align*}-\frac{1}{3}\end{align*}$ . Now, we need to find the $\begin{align*}y-\end{align*}$ intercept. Plug in 9 for $\begin{align*}x\end{align*}$ and -5 for $\begin{align*}y\end{align*}$ to solve for the new $\begin{align*}y-\end{align*}$ intercept $\begin{align*}(b)\end{align*}$ .
  ::回顾一条线的方程式是 y=mx+b, m 是 斜度, b 是 y- intercut 。 我们知道, 平行线的斜度相同, 因此线的斜度将为 - 13 。 现在, 我们需要找到 y- intercut 。 x 和 - 5 的插件为 y 的插件, 要解决新- intercut (b) 。

  播放段落

  $$\begin{align*}-5 & = -\frac{1}{3}(9)+b\\ -5 & = -3 + b\\ -2 & = b\end{align*}$$
  ::-513(9)+b-53+b-2=b

  播放段落

  The equation of parallel line is $\begin{align*}y = -\frac{1}{3}x-2\end{align*}$ .
  ::平行线的方程式是 y13x-2。

  播放段落

  Example 4
  ::例4

  播放段落

  Find the equation of the lines below and determine if they are parallel.
  ::查找下方线的方程式并确定它们是否平行 。

  

  播放段落

  The top line has a $\begin{align*}y-\end{align*}$ intercept of 1. From there, use “rise over run” to find the slope. From the $\begin{align*}y-\end{align*}$ intercept, if you go up 1 and over 2, you hit the line again, $\begin{align*}m = \frac{1}{2}\end{align*}$ . The equation is $\begin{align*}y=\frac{1}{2}x+1\end{align*}$ .
  ::顶端线有一个 y - intermission 的 y - intermission 1 。 从那里, 使用“ 向下 ” 来找到斜坡。 从 y - intermective , 如果您向上加 1 和 2, 您会再次按下线, m= 12。 方程式是 y= 12x+ 1 。

  播放段落

  For the second line, the $\begin{align*}y-\end{align*}$ intercept is -3. The “rise” is 1 and the “run” is 2 making the slope $\begin{align*}\frac{1}{2}\end{align*}$ . The equation of this line is $\begin{align*}y=\frac{1}{2}x-3\end{align*}$ .
  ::第二行,y - interview是 - 3。 " rise " 是1, " run " 是2, 使斜坡12。 这个线的方程式是y=12x-3。

  播放段落

  The lines are parallel because they have the same slope.
  ::线条是平行的,因为它们有相同的斜坡。

  播放段落

  Example 5
  ::例5

  播放段落

  Find the equation of the line that is parallel to the line through the point marked with a blue dot.
  ::查找与线平行的线的方程式,通过标有蓝色点的点。

  

  播放段落

  First, notice that the equation of the line is $\begin{align*}y=2x+6\end{align*}$ and the point is (2, -2). The parallel would have the same slope and pass through (2, -2).
  ::首先,请注意线的方程是y=2x+6,点是(2,2),平行线的斜度相同,通过(2,2)。

  播放段落

  $$\begin{align*}y & =2x+b\\ -2 & = 2(2) + b\\ -2 & = 4+b\\ -6 & = b\end{align*}$$
  ::y=2x+b-2=2(2)+b-2=4+b-6=b

  播放段落

  The equation of the parallel line is $\begin{align*}y=2x+-6\end{align*}$ .
  ::平行线的方程式是 y= 2x @% 6 。

  播放段落

  Review
  ::回顾

  播放段落

  Determine if each pair of lines are parallel. Then, graph each pair on the same set of axes.
  ::确定每条线是否平行。 然后, 将每对线都显示在同一组轴上 。

  播放段落
  1. $\begin{align*}y=4x-2\end{align*}$ and $\begin{align*}y=4x+5\end{align*}$
     ::y=4x-2和y=4x+5 y=4x-2和y=4x+5
  播放段落
  2. $\begin{align*}y=-x+5\end{align*}$ and $\begin{align*}y=x+1\end{align*}$
     ::yx+5和y=x+1 yx+5和y=x+1
  播放段落
  3. $\begin{align*}5x+2y=-4\end{align*}$ and $\begin{align*}5x+2y=8\end{align*}$
     ::5x+2y4和5x+2y=8
  播放段落
  4. $\begin{align*}x+y=6\end{align*}$ and $\begin{align*}4x+4y=-16\end{align*}$
     ::x+y=6和4x+4y16

  播放段落

  Determine the equation of the line that is parallel to the given line, through the given point.
  ::通过给定点确定与给定线平行的线条的方程。

  播放段落
  5. $\begin{align*}y=-5x+1; \ (-2, 3)\end{align*}$
     ::y5x+1;(-2,3)
  播放段落
  6. $\begin{align*}y=\frac{2}{3}x-2; \ (9, 1)\end{align*}$
     ::y=23x-2; (9,1)
  播放段落
  7. $\begin{align*}x-4y=12; \ (-16, -2)\end{align*}$
     ::x-4y=12; (- 16-2)
  播放段落
  8. $\begin{align*}3x+2y=10; \ (8, -11)\end{align*}$
     ::3x+2y=10; (8,-11)

  播放段落

  Find the equation of the two lines in each graph below. Then, determine if the two lines are parallel.
  ::在下图中查找两条线的方程式。 然后确定两条线是否平行 。

  9. 

  播放段落

  For the line and point below, find a parallel line, through the given point.
  ::对于下面的线和点,通过给定点找到一条平行线。

  10. 
  11. 
  12. 
  13. 

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  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 。