Course: 5.5 直线家庭 | The Way To Learn

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直线家庭数
Families of Lines
::直线家庭数

We can use the properties of parallel and perpendicular lines to write an equation of a line parallel or perpendicular to a given line. You might be given a line and a point, and asked to find the line that goes through the given point and is parallel or perpendicular to the given line. Here’s how to do this:
::我们可以使用平行线和直直线的属性来写出平行线或直线的等式。您可以给您一个线和点, 并请求您找到穿过给定点的线, 且该线与给定线平行或直线垂直。下面如何做 :

Find the of the given line from its equation. (You might need to re-write the equation in a form such as the slope-intercept form .)
::从等式中查找给定的线条。 (您可能需要用像斜坡界面的形式重新写入等式。 )

Find the slope of the parallel or perpendicular line—which is either the same as the slope you found in step 1 (if it’s parallel), or the negative reciprocal of the slope you found in step 1 (if it’s perpendicular).
::找到平行线或垂直线的斜坡 — — 与你在第1步(如果是平行线)中发现的斜坡相同,或者与你在第1步(如果是垂直线)中发现的斜坡负对等(如果是垂直线)相同。

Use the slope you found in step 2, along with the point you were given, to write an equation of the new line in slope-intercept form or point-slope form.
::使用第2步中发现的斜坡,加上给定的点,以斜坡截取形式或点斜坡形式写出新线的方程式。

Finding an Equation of a Line
::查找线条的公式

1. Find an equation of the line perpendicular to the line $\begin{aligned} y = - 3 x + 5 \end{aligned}$ that passes through the point (2, 6).
::1. 找到与横穿点(2、6)的 y3x+5 垂直的直线方程式。

The slope of the given line is -3, so the perpendicular line will have a slope of $\begin{aligned} \frac{1}{3} \end{aligned}$ .
::给定线的斜坡是 - 3, 因此垂直线的斜坡为13。

Now to find the equation of a line with slope $\begin{aligned} \frac{1}{3} \end{aligned}$ that passes through (2, 6):
::现在可以找到一条线的方程式,该线由13号斜坡通过(2,6):

Start with the slope-intercept form: $\begin{aligned} y = m x + b \end{aligned}$ .
::以 y=mx+b 的斜坡界面形式开始。

Plug in the slope: $\begin{aligned} y = \frac{1}{3} x + b \end{aligned}$ .
::斜坡中的插件:y=13x+b。

Plug in the point (2, 6) to find $\begin{aligned} b \end{aligned}$ : $\begin{aligned} 6 = \frac{1}{3} (2) + b \Rightarrow b = 6 - \frac{2}{3} \Rightarrow b = \frac{16}{3} \to 5 \frac{1}{3} \end{aligned}$ .
::插在点(2,6)中,找到 b: 6=13(2)+bb=6-23b=163513。

The equation of the line is $\begin{aligned} y = \frac{1}{3} x + 5 \frac{1}{3} \end{aligned}$ .
::线条等式=13x+513。

2. Find the equation of the line parallel to $\begin{aligned} 6 x - 5 y = 12 \end{aligned}$ that passes through the point (-5, -3).
::2. 找到与通过点(5,5,3)的 6x-5y=12 平行线的方程。

Rewrite the equation in slope-intercept form: $\begin{aligned} 6 x - 5 y = 12 \Rightarrow 5 y = 6 x - 12 \Rightarrow y = \frac{6}{5} x - \frac{12}{5} \end{aligned}$ .
::以斜坡界面形式重写方程式: 6x- 5y=125y=6x- 12 y=65x- 125。

The slope of the given line is $\begin{aligned} \frac{6}{5} \end{aligned}$ , so we are looking for a line with slope $\begin{aligned} \frac{6}{5} \end{aligned}$ that passes through the point (-5, -3).
::给定线的斜坡为65, 所以我们正在寻找一条斜坡65穿过点( 5, 3) 的线。

Start with the slope-intercept form: $\begin{aligned} y = m x + b \end{aligned}$ .
::以 y=mx+b 的斜坡界面形式开始。

Plug in the slope: $\begin{aligned} y = \frac{6}{5} x + b \end{aligned}$ .
::斜坡中的插件:y=65x+b。

Plug in the point (-5, -3): $\begin{aligned} n - 3 = \frac{6}{5} (- 5) + b \Rightarrow - 3 = - 6 + b \Rightarrow b = 3 \end{aligned}$
::点( 5, 3) 中的插头( 5, 3) : n- 3=65( 5) +b_ 3+b_ 6+b_b=3

The equation of the line is $\begin{aligned} y = \frac{6}{5} x + 3 \end{aligned}$ .
::等距 yy = 65x+3 的方程。

Investigate Families of Lines
::调查直线家属

A family of lines is a set of lines that have something in common with each other. Straight lines can belong to two types of families: one where the slope is the same and one where the $\begin{aligned} y - \end{aligned}$ intercept is the same.
::直线可以属于两种类型的家庭:一种是斜坡相同的家庭,另一种是y- intermission(y- interview)相同的家庭。

Family 1: Keep the slope unchanged and vary the $\begin{aligned} y - \end{aligned}$ intercept.
::家庭1:保持斜坡不变,并改变y- interviews。

The figure below shows the family of lines with equations of the form $\begin{aligned} y = - 2 x + b \end{aligned}$ :
::下图显示以 y2x+b 形式等式排列的线条组合:

All the lines have a slope of –2, but the value of $\begin{aligned} b \end{aligned}$ is different for each line.
::所有线条的斜度为-2, 但每条线的 b 值不同。

Notice that in such a family all the lines are parallel. All the lines look the same, except that they are shifted up and down the $\begin{aligned} y - \end{aligned}$ axis. As $\begin{aligned} b \end{aligned}$ gets larger the line rises on the $\begin{aligned} y - \end{aligned}$ axis, and as $\begin{aligned} b \end{aligned}$ gets smaller the line goes lower on the $\begin{aligned} y - \end{aligned}$ axis. This behavior is often called a vertical shift .
::请注意, 在这样的家庭里, 所有线条都是平行的。所有线条看起来都一样, 除了它们被向上和向下移动 y - 轴线。 b 越大, y - 轴线的上升, b 越小, y - 轴线的上升, y - 轴线的下降。这种行为通常被称为垂直变化。

Family 2: Keep the $\begin{aligned} y - \end{aligned}$ intercept unchanged and vary the slope.
::家庭 2: 保持y- interview 不变, 并改变斜坡。

The figure below shows the family of lines with equations of the form $\begin{aligned} y = m x + 2 \end{aligned}$ :
::下图显示以 y=mx+2 等式排列的线条组合: y=mx+2

All the lines have a $\begin{aligned} y - \end{aligned}$ intercept of two, but the slope is different for each line. The steeper lines have higher values of $\begin{aligned} m \end{aligned}$ .
::所有线条都有 y - 截取两个, 但每个线条的斜度不同。最陡的线条的数值是 m。

Writing an Equation of a Family of Lines
::书写一行系列的等式

Write the equation of the family of lines satisfying the given condition.
::写下符合特定条件的线条家族的方程式。

a) parallel to the $\begin{aligned} x - \end{aligned}$ axis
:a) 与x-轴平行

All lines parallel to the $\begin{aligned} x - \end{aligned}$ axis have a slope of zero; the $\begin{aligned} y - \end{aligned}$ intercept can be anything. So the family of lines is $\begin{aligned} y = 0 x + b \end{aligned}$ or just $\begin{aligned} y = b \end{aligned}$ .
::与 x - 轴相平行的所有线条的斜度为零; y - 界面可以是任何东西。因此, 线条的组合是 y= 0x+b 或 just y=b 。

b) through the point (0, -1)
:b) 通过点(0,-1)

All lines passing through the point (0, -1) have the same $\begin{aligned} y - \end{aligned}$ intercept, $\begin{aligned} b = - 1 \end{aligned}$ . The family of lines is: $\begin{aligned} y = m x - 1 \end{aligned}$ .
::穿过点( 0, - 1) 的所有线都有相同的y- intercut, b\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\y= mx-1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Y= mx-1\\\\\\\\\\\\\\\Y\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

c) perpendicular to $\begin{aligned} 2 x + 7 y - 9 = 0 \end{aligned}$
:c) 垂直至2x+7y-9=0

First we need to find the slope of the given line. Rewriting $\begin{aligned} 2 x + 7 y - 9 = 0 \end{aligned}$ in slope-intercept form, we get $\begin{aligned} y = - \frac{2}{7} x + \frac{9}{7} \end{aligned}$ . The slope of the line is $\begin{aligned} - \frac{2}{7} \end{aligned}$ , so we’re looking for the family of lines with slope $\begin{aligned} \frac{7}{2} \end{aligned}$ .
::首先,我们需要找到给定线的斜坡。重写 2x+7y- 9=0 的斜坡界面, 我们得到 y27x+97。线的斜坡是- 27, 所以我们正在寻找有斜坡72 的线条。

The family of lines is $\begin{aligned} y = \frac{7}{2} x + b \end{aligned}$ .
::线条的直线家族=72x+b。

d) parallel to $\begin{aligned} x + 4 y - 12 = 0 \end{aligned}$
::d) 与 x+4y-12=0平行

Rewrite $\begin{aligned} x + 4 y - 12 = 0 \end{aligned}$ in slope-intercept form: $\begin{aligned} y = - \frac{1}{4} x + 3 \end{aligned}$ . The slope is $\begin{aligned} - \frac{1}{4} \end{aligned}$ , so that’s also the slope of the family of lines we are looking for.
::以斜坡界面的形式重写 x+4y- 12=0 : y14x+3 。斜坡是 ~ 14, 也就是我们所寻找的线系的斜坡。

The family of lines is $\begin{aligned} y = - \frac{1}{4} x + b \end{aligned}$ .
::线条的家族 isy14x+b。

Example
::示例示例示例示例

Example 1
::例1

Find the equation of the line perpendicular to $\begin{aligned} x - 5 y = 15 \end{aligned}$ that passes through the point (-2, 5).
::查找横穿点(-2, 5)的直径至 x- 5y=15 的直线的方程。

Re-write the equation in slope-intercept form: $\begin{aligned} x - 5 y = 15 \Rightarrow - 5 y = - x + 15 \Rightarrow y = \frac{1}{5} x - 3 \end{aligned}$ .
::以斜坡界面形式重写方程式: x- 5y=15 @ @ @ 5y@ x+15_ y=15x- 3。

The slope of the given line is $\begin{aligned} \frac{1}{5} \end{aligned}$ , so we’re looking for a line with slope -5.
::给定线的斜坡是15, 所以我们正在寻找一条斜坡 - 5的斜坡。

Start with the slope-intercept form: $\begin{aligned} y = m x + b \end{aligned}$ .
::以 y=mx+b 的斜坡界面形式开始。

Plug in the slope: $\begin{aligned} y = - 5 x + b \end{aligned}$ .
::插入斜坡 : y5x+b 。

Plug in the point (-2, 5): $\begin{aligned} 5 = - 5 (- 2) + b \Rightarrow b = 5 - 10 \Rightarrow b = - 5 \end{aligned}$
::插插点 (-2, 2, 5): 55(-2) +bb=5- 10b5

The equation of the line is $\begin{aligned} y = - 5 x - 5 \end{aligned}$ .
::线的等式是#% 5x- 5 。

Review
::回顾

Find the equation of the line parallel to $\begin{aligned} 5 x - 2 y = 2 \end{aligned}$ that passes through point (3, -2).
::查找直线与通过点(3,2)的 5x-2y=2 平行的方程(3,2) 。

Find the equation of the line perpendicular to $\begin{aligned} y = - \frac{2}{5} x - 3 \end{aligned}$ that passes through point (2, 8).
::查找通过点(2,8)的y25x-3的垂直直线的方程。

Find the equation of the line parallel to $\begin{aligned} 7 y + 2 x - 10 = 0 \end{aligned}$ that passes through the point (2, 2).
::查找横穿点(2,2)的线的直线的等式,该直线与7y+2x-10=0平行。

Find the equation of the line perpendicular to $\begin{aligned} y + 5 = 3 (x - 2) \end{aligned}$ that passes through the point (6, 2).
::查找横穿点(6,2)的y+5=3(x-2)垂直线的方程。

Line $\begin{aligned} S \end{aligned}$ passes through the points (2, 3) and (4, 7). Line $\begin{aligned} T \end{aligned}$ passes through the point (2, 5). If Lines $\begin{aligned} S \end{aligned}$ and $\begin{aligned} T \end{aligned}$ are parallel, name one more point on line $\begin{aligned} T \end{aligned}$ . ( Hint: you don’t need to find the slope of either line.)
::S线通过点(2、3和4、7),T线通过点(2、5),如果S线和T线是平行的,请在T线上再指出一个点(提示:不需要找到两线的斜坡)。

Lines $\begin{aligned} P \end{aligned}$ and $\begin{aligned} Q \end{aligned}$ both pass through (-1, 5). Line $\begin{aligned} P \end{aligned}$ also passes through (-3, -1). If $\begin{aligned} P \end{aligned}$ and $\begin{aligned} Q \end{aligned}$ are perpendicular, name one more point on line $\begin{aligned} Q \end{aligned}$ . (This time you will have to find the slopes of both lines.)
::P线和Q线都通过(-1, 5),P线也通过(-3, -1)。如果P和Q是垂直的,请在Q线上再指出一个点。 (这次你必须找到两线的斜坡。 )

Write the equation of the family of lines satisfying the given condition.

All lines that pass through point (0, 4).
::所有通过点(0, 4)的线条。

All lines that are perpendicular to $\begin{aligned} 4 x + 3 y - 1 = 0 \end{aligned}$ .
::所有直径为 4x+3y- 1=0 的直线。

All lines that are parallel to $\begin{aligned} y - 3 = 4 x + 2 \end{aligned}$ .
::与 y- 3= 4x+2 平行的所有线条。

All lines that pass through the point (0, -1).
::穿过点( 0, - 1) 的所有线条。

::写入符合给定条件的线条组的方程式。所有通过点( 0, 4) 的线条,所有直线都垂直到 4x+3y-1=0。所有直线都与 y-3=4x+2 平行。所有直线都通过点( 0, - 1) 。

Name two lines that pass through the point (3, -1) and are perpendicular to each other.
::标出通过点的两行(3,-1),并且相互垂直。

Name two lines that are each perpendicular to $\begin{aligned} y = - 4 x - 2 \end{aligned}$ . What is the relationship of those two lines to each other?
::列出两个与 y4x-2 垂直的线条。这两条线之间有什么关系?

Name two perpendicular lines that both pass through the point (3, -2). Then name a line parallel to one of them that passes through the point (-2, 5).
::列出通过点(3, - 2)的两条垂直直径线,然后列出一条与通过点(-2, 5)的直线平行的直线。

Review (Answers)
::回顾(答复)

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。

Section outline

General

直线家庭数

Families of Lines ::直线家庭数

Finding an Equation of a Line ::查找线条的公式

Investigate Families of Lines ::调查直线家属

Writing an Equation of a Family of Lines ::书写一行系列的等式

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Families of Lines
::直线家庭数

Finding an Equation of a Line
::查找线条的公式

Investigate Families of Lines
::调查直线家属

Writing an Equation of a Family of Lines
::书写一行系列的等式

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)