Course: 5.2 线性等式形式

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线性方形
Forms of Linear Equations
::线性方形

You’ve already encountered another useful form for writing linear equations: standard form . An equation in standard form is written $\begin{aligned} a x + b y = c \end{aligned}$ , where $\begin{aligned} a, b \end{aligned}$ , and $\begin{aligned} c \end{aligned}$ are all integers and $\begin{aligned} a \end{aligned}$ is positive. (Note that the $\begin{aligned} b \end{aligned}$ in the standard form is different than the $\begin{aligned} b \end{aligned}$ in the slope-intercept form .)
::您已经遇到另一个用于写入线性方程式的有用格式: 标准格式。标准格式的方程式是书写轴+by=c, 其中 a、 b 和 c 是全部整数, a 是正数。 (注意标准格式中的 b 与斜度- 界面中的 b 不同。)

One useful thing about standard form is that it allows us to write equations for vertical lines, which we can’t do in slope-intercept form.
::标准格式的一个有用之处是,它允许我们为垂直线写方程式,而我们无法用斜坡拦截形式这样做。

For example, let’s look at the line that passes through points (2, 6) and (2, 9). How would we find an equation for that line in slope-intercept form?
::例如,让我们看看通过点(2、6和2、9)的线条。我们如何用斜坡界面找到这条线的方程式?

First we’d need to find the : $\begin{aligned} m = \frac{9 - 6}{0 - 0} = \frac{3}{0} \end{aligned}$ . But that slope is undefined because we can’t divide by zero. And if we can’t find the slope, we can’t use point-slope form either.
::首先,我们需要找到:m=9-60-0=30。但是,这个斜坡没有定义,因为我们无法除以零。如果我们找不到斜坡,我们也不能使用斜坡形式。

If we just graph the line, we can see that $\begin{aligned} x \end{aligned}$ equals 2 no matter what $\begin{aligned} y \end{aligned}$ is. There’s no way to express that in slope- intercept or point-slope form, but in standard form we can just say that $\begin{aligned} x + 0 y = 2 \end{aligned}$ , or simply $\begin{aligned} x = 2 \end{aligned}$ .
::如果我们只绘制线条图, 我们可以看到 x 等于 2 , 不管是什么 y 是什么。无法用斜坡界面或点窗体表达这一点, 但标准格式我们只能说 x+0y=2, 或者简单地说 x=2 。

Converting to Standard Form
::转换为标准表格

To convert an equation from another form to standard form, all you need to do is rewrite the equation so that all the variables are on one side of the equation and the coefficient of $\begin{aligned} x \end{aligned}$ is not negative.
::要将方程式从另一种形式转换为标准形式,你只需要重写方程式,使所有变量都在方程式的一边,x的系数不是负数。

Rewrite the following equations in standard form:
::以标准格式重写以下方程式:

We need to rewrite each equation so that all the variables are on one side and the coefficient of $\begin{aligned} x \end{aligned}$ is not negative.
::我们需要重写每个方程, 以便所有变量都站在一边, x 系数不是负数。

a) $\begin{aligned} y = 5 x - 7 \end{aligned}$
:a)y=5x-7

$\begin{aligned} y = 5 x - 7 \end{aligned}$
::y=5x-7 y=5x-7

Subtract $\begin{aligned} y \end{aligned}$ from both sides to get $\begin{aligned} 0 = 5 x - y - 7 \end{aligned}$ .
::从两边减去y, 以获得 0= 5x- y- 7 。

Add 7 to both sides to get $\begin{aligned} 7 = 5 x - y \end{aligned}$ .
::双方加7,7=5xy

Flip the equation around to put it in standard form: $\begin{aligned} 5 x - y = 7 \end{aligned}$ .
::翻转方程式以将其设置为标准格式: 5x-y=7。

b) $\begin{aligned} y - 2 = - 3 (x + 3) \end{aligned}$
:b)y-23(x+3)

$\begin{aligned} y - 2 = - 3 (x + 3) \end{aligned}$
::y-23(x+3)

Distribute the –3 on the right-hand-side to get $\begin{aligned} y - 2 = - 3 x - 9 \end{aligned}$ .
::右手边的 -3 分配到右手边去获得 y-23x- 9 。

Add $\begin{aligned} 3 x \end{aligned}$ to both sides to get $\begin{aligned} y + 3 x - 2 = - 9 \end{aligned}$ .
::向两边添加 3x 来获取 y+3x-2\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Add 2 to both sides to get $\begin{aligned} y + 3 x = - 7 \end{aligned}$ . Flip that around to get $\begin{aligned} 3 x + y = - 7 \end{aligned}$ .
::向两边加2 以获得 y+3x+7 。翻转它以获得 3x+y+7 。

c) $\begin{aligned} y = \frac{2}{3} x + \frac{1}{2} \end{aligned}$
::c) y=23x+12

$\begin{aligned} y = \frac{2}{3} x + \frac{1}{2} \end{aligned}$
::y=23x+12 y=23x+12

Find the common denominator for all terms in the equation – in this case that would be 6.
::找到方程式中所有条件的共同分母 — — 在本案中是6个。

Multiply all terms in the equation by 6: $\begin{aligned} 6 (y = \frac{2}{3} x + \frac{1}{2}) \Rightarrow 6 y = 4 x + 3 \end{aligned}$
::将方程中的所有条件乘以 6 : 6(y= 23x+12+12)\\\ 6y= 4x+3

Subtract $\begin{aligned} 6 y \end{aligned}$ from both sides: $\begin{aligned} 0 = 4 x - 6 y + 3 \end{aligned}$
::双方减6Y:0=4x-6y+3

Subtract 3 from both sides: $\begin{aligned} - 3 = 4 x - 6 y \end{aligned}$
::双方减3:-3=4x-6y

The equation in standard form is $\begin{aligned} 4 x - 6 y = - 3 \end{aligned}$ .
::标准方程式的方程式是 4x-6y3。

Graphing Equations in Standard Form
::标准表格中的图形等号

When an equation is in slope-intercept form or point-slope form, you can tell right away what the slope is. How do you find the slope when an equation is in standard form?
::当一个方程式以斜坡截面形式或点斜坡形式出现时,您可以立即辨别斜坡是什么。当一个方程式以标准形式出现时,您如何找到斜坡 ?

Well, you could rewrite the equation in slope-intercept form and read off the slope. But there’s an even easier way. Let’s look at what happens when we rewrite an equation in standard form.
::你可以以斜坡界面的形式重写方程式,从斜坡上读取。但有一个更加简单的方法。让我们看看我们以标准格式重写方程式时会发生什么。

Starting with the equation $\begin{aligned} a x + b y = c \end{aligned}$ , we would subtract $\begin{aligned} a x \end{aligned}$ from both sides to get $\begin{aligned} b y = - a x + c \end{aligned}$ . Then we would divide all terms by $\begin{aligned} b \end{aligned}$ and end up with $\begin{aligned} y = - \frac{a}{b} x + \frac{c}{b} \end{aligned}$ .
::从方程式 ax+by=c 开始,我们将从两边减去斧头, 以通过 {ax+c 获得。然后我们将所有条件除以 b, 最后加上 yabx+cb 。

That means that the slope is $\begin{aligned} - \frac{a}{b} \end{aligned}$ and the $\begin{aligned} y - \end{aligned}$ intercept is $\begin{aligned} \frac{c}{b} \end{aligned}$ . So next time we look at an equation in standard form, we don’t have to rewrite it to find the slope; we know the slope is just $\begin{aligned} - \frac{a}{b} \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ are the coefficients of $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ in the equation.
::这意味着斜坡是-ab, y- intercutism是 cb。所以下次我们看标准方程式的方程式时, 我们不必重写以找到斜坡; 我们知道斜坡只是-ab, 其中a和b是方程式中的 x 和 y 的系数。

Finding the Slope and $\begin{aligned} y - \end{aligned}$ Intercept
::查找斜坡和y - 截取

Find the slope and the $\begin{aligned} y - \end{aligned}$ intercept of the following equations written in standard form.
::查找以标准格式书写的下列方程式的斜度和 Y - interview 。

a) $\begin{aligned} 3 x + 5 y = 6 \end{aligned}$
::a) 3x+5y=6

$\begin{aligned} a = 3, b = 5 \end{aligned}$ , and $\begin{aligned} c = 6 \end{aligned}$ , so the slope is $\begin{aligned} - \frac{a}{b} = - \frac{3}{5} \end{aligned}$ , and the $\begin{aligned} y - \end{aligned}$ intercept is $\begin{aligned} \frac{c}{b} = \frac{6}{5} \end{aligned}$ .
::a=3,b=5,和c=6,所以斜坡是-ab35,y-interview是 cb=65。

b) $\begin{aligned} 2 x - 3 y = - 8 \end{aligned}$
:b) 2x-3y8

$\begin{aligned} a = 2, b = - 3 \end{aligned}$ , and $\begin{aligned} c = - 8 \end{aligned}$ , so the slope is $\begin{aligned} - \frac{a}{b} = \frac{2}{3} \end{aligned}$ , and the $\begin{aligned} y - \end{aligned}$ intercept is $\begin{aligned} \frac{c}{b} = \frac{8}{3} \end{aligned}$ .
::a=2,b3,和c8,所以斜坡是-ab=23,y-interview是cb=83。

c) $\begin{aligned} x - 5 y = 10 \end{aligned}$
:c) x-5y=10

$\begin{aligned} a = 1, b = - 5 \end{aligned}$ , and $\begin{aligned} c = 10 \end{aligned}$ , so the slope is $\begin{aligned} - \frac{a}{b} = \frac{1}{5} \end{aligned}$ , and the $\begin{aligned} y - \end{aligned}$ intercept is $\begin{aligned} \frac{c}{b} = \frac{10}{- 5} = - 2 \end{aligned}$ .
::a=1,b5,和c=10,所以斜坡是-ab=15,y - interview是 cb=10-52。

Once we’ve found the slope and $\begin{aligned} y - \end{aligned}$ intercept of an equation in standard form, we can graph it easily. But if we start with a graph, how do we find an equation of that line in standard form?
::一旦我们找到了标准形式的方程式的斜坡和 Y- 界面, 我们可以很容易地用图表来图解它。但如果我们从图表开始, 我们如何在标准格式中找到该线的方程式 ?

First, remember that we can also use the cover-up method to graph an equation in standard form, by finding the intercepts of the line. For example, let’s graph the line given by the equation $\begin{aligned} 3 x - 2 y = 6 \end{aligned}$ .
::首先, 请记住, 我们也可以使用隐蔽方法, 以标准格式绘制方程式, 找到线的截取。例如, 让我们来绘制方程式 3x-2y= 6 给出的线条。

To find the $\begin{aligned} x - \end{aligned}$ intercept, cover up the $\begin{aligned} y \end{aligned}$ term (remember, the $\begin{aligned} x - \end{aligned}$ intercept is where $\begin{aligned} y = 0 \end{aligned}$ ):
::要找到 x - intercut, 覆盖 Y 术语( 记住, x - intercut is where y=0) :

$\begin{aligned} 3 x = 6 \Rightarrow x = 2 \end{aligned}$
::3x=6x=2

The $\begin{aligned} x - \end{aligned}$ intercept is (2, 0).
::x - 拦截为(2 0) 。

To find the $\begin{aligned} y - \end{aligned}$ intercept, cover up the $\begin{aligned} x \end{aligned}$ term (remember, the $\begin{aligned} y - \end{aligned}$ intercept is where $\begin{aligned} x = 0) \end{aligned}$ :
::要找到 y - 界面, 覆盖 x 术语( 记住, y - 界面是 x= 0 的位置 ) :

$\begin{aligned} - 2 y = 6 \Rightarrow y = - 3 \end{aligned}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The $\begin{aligned} y - \end{aligned}$ intercept is (0, -3).
::y- 拦截是 (0, - 3) 。

We plot the intercepts and draw a line through them that extends in both directions:
::我们绘制拦截线,通过它们划出一条线,从两个方向延伸:

Now we want to apply this process in reverse—to start with the graph of the line and write the equation of the line in standard form.
::现在,我们想反向应用这个过程——从线条的图形开始,用标准格式写出线条的方程式。

Finding and Re-Writing the Equation of a Line
::查找和重新写入线条的公式

Find the equation of each line and write it in standard form.
::查找每行的方程, 并以标准格式写入。

a)
::a) (a)

We see that the $\begin{aligned} x - \end{aligned}$ intercept is $\begin{aligned} (3, 0) \Rightarrow x = 3 \end{aligned}$ and the $\begin{aligned} y - \end{aligned}$ intercept is $\begin{aligned} (0, - 4) \Rightarrow y = - 4 \end{aligned}$
::我们看到 X - 截取是 (3,0) x= 3, y - 截取是 (0, - 4) 4

We saw that in standard form $\begin{aligned} a x + b y = c \end{aligned}$ : if we “cover up” the $\begin{aligned} y \end{aligned}$ term, we get $\begin{aligned} a x = c \end{aligned}$ , and if we “cover up” the $\begin{aligned} x \end{aligned}$ term, we get $\begin{aligned} b y = c \end{aligned}$ .
::我们用标准格式x+by=c看到:如果我们“掩盖”y 术语,我们就会得到ax=c,如果我们“掩盖”x 术语,我们就会得到 by=c。

So we need to find values for $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ so that we can plug in 3 for $\begin{aligned} x \end{aligned}$ and -4 for $\begin{aligned} y \end{aligned}$ and get the same value for $\begin{aligned} c \end{aligned}$ in both cases. This is like finding the least common multiple of the $\begin{aligned} x - \end{aligned}$ and $\begin{aligned} y - \end{aligned}$ intercepts.
::因此我们需要找到 a 和 b 的值, 这样我们就可以在 x 和 - 4 的 y 中插入 3 和 - 4 的 y 中插入 3 , 在两种情况下 c 的值相同。这就像找到 x 和 y - interview 中最不常见的倍数一样。

In this case, we see that multiplying $\begin{aligned} x = 3 \end{aligned}$ by 4 and multiplying $\begin{aligned} y = - 4 \end{aligned}$ by –3 gives the same result:
::在这种情况下,我们看到 x=3 乘以 4 乘以 - 3 得出同样的结果 :

$\begin{aligned} (x = 3) \times 4 \Rightarrow 4 x = 12 and (y = - 4) \times (- 3) \Rightarrow - 3 y = 12 \end{aligned}$

:x=3) x44x=12and(y4) x(-3) 3y=12)

Therefore , $\begin{aligned} a = 4, b = - 3 \end{aligned}$ and $\begin{aligned} c = 12 \end{aligned}$ and the equation in standard form is $\begin{aligned} 4 x - 3 y = 12 \end{aligned}$ .
::因此,a=4,b3和c=12,标准方程式的方程式为4x-3y=12。

b)
:b) b)

We see that the $\begin{aligned} x - \end{aligned}$ intercept is $\begin{aligned} (3, 0) \Rightarrow x = 3 \end{aligned}$ and the $\begin{aligned} y - \end{aligned}$ intercept is $\begin{aligned} (0, 3) \Rightarrow y = 3 \end{aligned}$
::我们看到 X - 截取是 (3,0) x= 3, y - 截取是 (0,3) y= 3

The values of the intercept equations are already the same, so $\begin{aligned} a = 1, b = 1 \end{aligned}$ and $\begin{aligned} c = 3 \end{aligned}$ . The equation in standard form is $\begin{aligned} x + y = 3 \end{aligned}$ .
::截取方程式的值已经相同, 所以 a=1, b=1 和 c=3 。标准格式为 exx+y=3 的方程式。

c)
:c) c)

We see that the $\begin{aligned} x - \end{aligned}$ intercept is $\begin{aligned} (\frac{3}{2}, 0) \Rightarrow x = \frac{3}{2} \end{aligned}$ and the $\begin{aligned} y - \end{aligned}$ intercept is $\begin{aligned} (0, 4) \Rightarrow y = 4 \end{aligned}$
::我们看到 X - 界面是 (32,0) x= 32, y - 界面是 (0,4) y= 4

Let’s multiply the $\begin{aligned} x - \end{aligned}$ intercept equation by $\begin{aligned} 2 \Rightarrow 2 x = 3 \end{aligned}$
::让我们将 x 乘以 2\\\ 2x=3 乘以 X - 截取方程式

Then we see we can multiply the $\begin{aligned} x - \end{aligned}$ intercept again by 4 and the $\begin{aligned} y - \end{aligned}$ intercept by 3, so we end up with $\begin{aligned} 8 x = 12 \end{aligned}$ and $\begin{aligned} 3 y = 12 \end{aligned}$ .
::然后我们可以看到, x - 拦截再次乘以 4 和y - 拦截乘以 3, 所以我们最终会乘以 8x= 12 和 3y= 12 。

The equation in standard form is $\begin{aligned} 8 x + 3 y = 12 \end{aligned}$ .
::标准格式的方程式是 8x+3y=12 。

Examples
::实例

Find the slope and the $\begin{aligned} y - \end{aligned}$ intercept of the following equations written in standard form.
::查找以标准格式书写的下列方程式的斜度和 Y - interview 。

Example 1
::例1

$\begin{aligned} 10 x + 2 y = 5 \end{aligned}$
::10x+2y=5

$\begin{aligned} a = 10, b = 2 \end{aligned}$ , and $\begin{aligned} c = 5 \end{aligned}$ , so the slope is $\begin{aligned} - \frac{a}{b} = - \frac{10}{2} = - 5 \end{aligned}$ , and the $\begin{aligned} y - \end{aligned}$ intercept is $\begin{aligned} \frac{5}{2} = 2.5 \end{aligned}$ .
::a=10,b=2,c=5,所以斜坡是-ab102.5,y-interview是52=2.5。

Example 2
::例2

$\begin{aligned} 21 x - 3 y = - 9 \end{aligned}$
::21 - 3y9

$\begin{aligned} a = 21, b = - 3 \end{aligned}$ , and $\begin{aligned} c = - 9 \end{aligned}$ , so the slope is $\begin{aligned} - \frac{a}{b} = - \frac{21}{- 3} = 7 \end{aligned}$ , and the $\begin{aligned} y - \end{aligned}$ intercept is $\begin{aligned} \frac{c}{b} = \frac{- 9}{- 3} = 3 \end{aligned}$ .
::a=21,b3,和c9, 所以斜坡是-ab21-3=7, y-interview是 cb9-3=3。

Review
::回顾

For 1-6, rewrite the following equations in standard form.
::1-6, 以标准格式重写以下方程式。

$\begin{aligned} y = 3 x - 8 \end{aligned}$
::y=3x-8

$\begin{aligned} y - 7 = - 5 (x - 12) \end{aligned}$
::y-7+5(x-12)

$\begin{aligned} 2 y = 6 x + 9 \end{aligned}$
::2y=6x+9

$\begin{aligned} y = \frac{9}{4} x + \frac{1}{4} \end{aligned}$
::y=94x+14 y=94x+14

$\begin{aligned} y + \frac{3}{5} = \frac{2}{3} (x - 2) \end{aligned}$
::y+35=23(x-2)

$\begin{aligned} 3 y + 5 = 4 (x - 9) \end{aligned}$
::3y+5=4(x-9)

For 7-12, find the slope and $\begin{aligned} y - \end{aligned}$ intercept of the following lines.
::7-12, 找到以下线条的斜坡和 y- intercide 。

$\begin{aligned} 5 x - 2 y = 15 \end{aligned}$
::5x-2yy=15

$\begin{aligned} 3 x + 6 y = 25 \end{aligned}$
::3x+6y=25

$\begin{aligned} x - 8 y = 12 \end{aligned}$
::x-8y=12 x-8y=12

$\begin{aligned} 3 x - 7 y = 20 \end{aligned}$
::3x-7y=20

$\begin{aligned} 9 x - 9 y = 4 \end{aligned}$
::9-9y=4

$\begin{aligned} 6 x + y = 3 \end{aligned}$
::6x+y=3

For 13-14, find the equation of each line and write it in standard form.
::13-14, 找到每一行的方程, 并以标准格式写成。

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

线性方形

Forms of Linear Equations ::线性方形

Graphing Equations in Standard Form ::标准表格中的图形等号

Finding the Slope and y − Intercept ::查找斜坡和y - 截取

Finding and Re-Writing the Equation of a Line ::查找和重新写入线条的公式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)

Forms of Linear Equations
::线性方形

Graphing Equations in Standard Form
::标准表格中的图形等号

Finding the Slope and $\begin{aligned} y - \end{aligned}$ Intercept
::查找斜坡和y - 截取

Finding and Re-Writing the Equation of a Line
::查找和重新写入线条的公式

Examples
::实例

Example 1
::例1

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)