Course: 9.9 使用斜坡拦截表

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使用斜坡截取表
Alyssa, a grade eight student, was looking through the school newsletter when she saw an ad placed by a local farmer looking for students to pick strawberries. The farmer has offered to pay the students a daily rate of $15.00 plus $3.00 for each quart basket of strawberries they pick.
::Alyssa是八年级学生,当时她看到当地农民张贴广告,要求学生采摘草莓,她正在翻阅学校通讯。农民提出每天向学生支付15美元加3.00美元,每采摘一夸特篮草莓。

Alyssa is interested in applying for a job but she would like to model the information with some type of a linear equation so she can figure out how much money she could make picking strawberries in one day. Alyssa is confident she can pick 50 quarts of berries in one day.
::Alyssa有兴趣申请一份工作,但她想用某种线性方程来模拟这些信息,这样她就可以知道她一天能挣多少钱来摘草莓。 Alyssa有信心在一天内可以摘50夸的浆果。

How can Alyssa figure this out?
::Alyssa怎么弄清楚这件事?

In this concept, you will learn to use slope-intercept form .
::在此概念中,您将学会使用斜坡拦截形式。

Slope Intercept Form
::斜坡截截表

A linear equation can be expressed in standard form as $\begin{align*}Ax+By=C\end{align*}$ such that $\begin{align*}A \neq 0\end{align*}$ and $\begin{align*}B \neq 0\end{align*}$ . When the equation is written in this form the $\begin{align*}x\end{align*}$ - and $\begin{align*}y\end{align*}$ - intercepts are easily found using the formulas:
::线性方程式可以以 Ax+By=C 的标准格式表示,例如 A0 和 B0。当方程式以这种形式写成时,使用公式很容易找到 x 和 y 界面:

The $\begin{align*}x\text{-intercept} = \tfrac{C}{A} \end{align*}$ and $\begin{align*}y\text{-intercept}=\tfrac{C}{B}\end{align*}$ . To find the of the line using the equation written in standard form requires using algebra to solve the equation for the variable ‘ $\begin{align*}y\end{align*}$ ’.
::x- 截取=CA 和 y- 截取=CB 。要用标准格式书写方程式找到线条, 需要使用代数来解析变量“ y” 的方程。

Linear equations can also be expressed in slope-intercept form as $\begin{align*}y=mx+b\end{align*}$ such that ‘ $\begin{align*}m\end{align*}$ ’ is the slope of the line and ‘ $\begin{align*}b\end{align*}$ ’ is the $\begin{align*}y\end{align*}$ - intercept of the line.
::线性方程式也可以以 y=mx+b 的斜坡界面表达为 y=mx+b , 使 'm ' 是线的斜坡, `b ' 是线的 Y 界面。

Let’s look at an example:
::让我们举个例子:

Graph the linear equation $\begin{align*}y=3x+2\end{align*}$ .
::绘制线性方程 y= 3x+2 。

The graph of the line crosses the $\begin{align*}y\end{align*}$ -axis at the point $\begin{align*}(0, 2)\end{align*}$ . The slope of the line is one unit to the right (change in $\begin{align*}x\end{align*}$ ) and up three units (change in $\begin{align*}y\end{align*}$ ). The value of ‘ $\begin{align*}m\end{align*}$ ’ is $\begin{align*}\frac{3}{1}\end{align*}$ .
::线条图在点( 0, 2) 横过 Y 轴。线的斜坡是右方的一个单位( x 的改变) , 向上是三个单位( y 的改变)。 ` m ' 的值是 31 。

A linear equation written in standard form can be changed to slope-intercept form using algebra.
::以标准格式写成的线性方程式可以使用代数转换成斜度截取形式。

Given the equation $\begin{align*}3x+2y = 12\end{align*}$ written in standard form, rewrite it in slope-intercept form.
::3x+2y=12 方程式以标准格式写成,请以斜坡界面格式重写。

$\begin{aligned} 3 x + 2 y = 12 \end{aligned}$
$\begin{align*}3x + 2y =12\end{align*}$
::3x+2y=12

First, subtract $\begin{align*}3x\end{align*}$ from both sides of the equation to isolate the ‘ $\begin{align*}y\end{align*}$ ’ term .
::首先,从方程的两侧减去3x,以孤立“y”一词。

$\begin{aligned} \begin{array}{rcl} 3 x + 2 y & = & 12 \\ 3 x - 3 x + 2 y & = & 12 - 3 x \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 3x+2y &=& 12 \\ 3x - 3x + 2y &=& 12 - 3x \end{array}\end{align*}$
::3x+2y=123x-3x+2y=12-3x

Next, simplify both sides of the equation.
::接下来,简化方程式的两边

$\begin{aligned} \begin{array}{rcl} 3 x - 3 x + 2 y & = & 12 - 3 x \\ 2 y & = & 12 - 3 x \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 3x -3x + 2y &=& 12 - 3x \\ 2y &=& 12 - 3x \end{array}\end{align*}$
::3x-3x+2y=12-3x2y=12-3x2y=12-3xx

Next, rewrite the right side of the equation to correspond to $\begin{align*}y=mx+b\end{align*}$ .
::下一位, 重写方程的右侧以对应 y=mx+b 。

$\begin{aligned} \begin{array}{rcl} 2 y & = & 12 - 3 x \\ 2 y & = & - 3 x + 12 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 2y &=& 12 -3x \\ 2y &=& -3x +12 \end{array}\end{align*}$
::2y=12- 3x2y%3x+12

Then, divide both sides of the equation by ‘2’ to solve for the variable ‘ $\begin{align*}y\end{align*}$ ’.
::然后将方程两边除以 '2 ' ' ' ' ' , 解决变量`y ' 。

$\begin{aligned} \begin{array}{rcl} 2 y & = & - 3 x + 12 \\ \frac{\overset{1}{2} y}{2} & = & \frac{- 3 x}{2} + \frac{\overset{6}{12}}{2} \\ y & = & - \frac{3}{2} x + 6 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 2y &=& -3x +12 \\ \frac{\overset{1}{\cancel{2}}y}{{\cancel{2}}} &=& \frac{-3x}{2} + \frac{\overset{6} {\cancel{12}}}{{\cancel{2}}} \\ y &=& - \frac{3}{2} x + 6 \end{array}\end{align*}$
::2 y3x+1221y23x2+1262y32x+6

The answer is $\begin{align*}y=-\frac{3}{2} x +6\end{align*}$ .
::答案是y32x+6。

For the following equation, identify the slope and the $\begin{align*}y\end{align*}$ -intercept of the line:
::对于以下方程,标明斜坡和横线的 Y 插口:

$\begin{aligned} y = - \frac{3}{2} x + 6 \end{aligned}$
$\begin{align*}y=-\frac{3}{2} x +6\end{align*}$
::y32x+6

The equation is in the form
::方程式在形式上

$\begin{aligned} y = m x + b \end{aligned}$
$\begin{align*}y = mx +b\end{align*}$
::y=mx+b y=mx+b

The value of the slope ( $\begin{align*}m\end{align*}$ ) is $\begin{align*}-\frac{3}{2}\end{align*}$ and the $\begin{align*}y\end{align*}$ -intercept ( $\begin{align*}b\end{align*}$ ) is $\begin{align*}(0, 6)\end{align*}$ .
::斜坡(m) 值为-32,y 接口(b) 值为(0,6) 。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Alyssa and the strawberries. She needs to figure out how much money she can make picking 50 quarts of strawberries in one day. How can she do this?
::早些时候,你被问及Alyssa和草莓的问题。她需要弄清楚她一天能挣多少钱来摘50夸特的草莓。她怎么能这样做呢?

Alyssa can write an equation in slope intercept form to represent the information she knows.
::Alyssa可以用斜坡截击形式写一个方程式,以代表她所知道的信息。

First, write down the information given in the ad.
::首先,写下广告中提供的信息。

$3.00 for each quart of berries picked plus $15.00 per day.
::每采果子一夸特,每采果子3美元,每天15美元。

Next, write down the slope-intercept form of an equation.
::接下来,写下一个方程的斜坡拦截形式。

$\begin{aligned} y = m x + b \end{aligned}$
$\begin{align*}y = {\color{blue}m}x +{\color{red}b}\end{align*}$
::y=mx+b y=mx+b

Next, for the given information, what do the letters represent?
::其次,对于给定的信息,字母代表什么?

$\begin{align*}y\end{align*}$ = amount of money earned
::y = 挣得的金钱数额

$\begin{align*}m\end{align*}$ = amount paid for each quart of berries picked
::m = 每采摘果浆一夸特支付的数额

$\begin{align*}x\end{align*}$ = number of quarts of berries picked
::x = 采摘浆果的夸特数量

$\begin{align*}b\end{align*}$ = amount paid for each day worked
::b = 每天工作支付的数额。

Next, fill in the known values into the equation.
::接下来,填入方程中的已知值。

$\begin{aligned} \begin{array}{rcl} y & = & m x + b \\ y & = & 3 x + 15 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} y &=& {\color{blue}m}x + {\color{red}b} \\ y &=& {\color{blue}3}x +{\color{red}15} \end{array}\end{align*}$
::y=mx+by=3x+15 y=mx+by=3x+15

Then, using the equation fill in 50 for ‘ $\begin{align*}x\end{align*}$ ’ and simplify to calculate the total amount of money.
::然后,用50的方程式填充 'x' 并简化计算金额总额。

$\begin{aligned} \begin{array}{rcl} y & = & 3 x + 15 \\ y & = & 3 (50) + 15 \\ y & = & 150 + 15 \\ y & = & 165 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} y &=& {\color{blue}3}x + {\color{red}15} \\ y &=& {\color{blue}3} (50) + {\color{red}15} \\ y &=& 150 + 15 \\ y &=& 165 \end{array}\end{align*}$
::y=3x+15y=3(50)+15y=150+15y=165

The answer is 165.
::答案是165

Alyssa can earn $165.00 in one day picking strawberries.
::Alyssa一天就能赚到165美元摘草莓

Example 2
::例2

Write the following equation in slope-intercept form and state the slope and the $\begin{align*}y\end{align*}$ -intercept of the line.
::以斜坡截取形式写下以下方程式,并标明斜坡和横线的 Y 截取。

$\begin{aligned} - 18 x + 6 y = 12 \end{aligned}$
$\begin{align*}-18x+6y =12\end{align*}$
::- 18x+6y=12

First, add $\begin{align*}18x\end{align*}$ to both sides of the equation and simplify.
::首先,在方程的两侧加上18x并简化。

$\begin{aligned} \begin{array}{rcl} - 18 x + 6 y & = & 12 \\ - 18 x + 18 x + 6 y & = & 12 + 18 x \\ 6 y & = & 12 + 18 x \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} -18 x + 6y &=& 12 \\ -18 x+18x+6y &=& 12 + 18x \\ 6y &=& 12 + 18x \end{array}\end{align*}$
::- 18x+6y=12-18x+18x+6y=12+18x6y=12+18x6y=12+18x

Next, divide both sides of the equation by ‘6’ to solve for ‘ $\begin{align*}y\end{align*}$ ’.
::接下来,将方程式的两边除以 ' 6 ' ,

$\begin{aligned} \begin{array}{rcl} 6 y & = & 12 + 18 x \\ \frac{\overset{1}{6} y}{6} & = & \frac{\overset{2}{12}}{6} + \frac{\overset{3}{18} x}{6} \\ y & = & 2 + 3 x \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 6y &=& 12 +18x \\ \frac{\overset{1}{\cancel{6}}y}{{\cancel{6}}} &=& \frac{\overset{2}{\cancel{12}}}{{\cancel{6}}} + \frac{\overset{3} {\cancel{18}}x}{{\cancel{6}}} \\ y &=& 2+3x \end{array}\end{align*}$
::6y=12+18x61y6=1226+183x6y=2+3x

Then, express the equation in slope-intercept form.
::然后用斜坡界面表达方程式

$\begin{aligned} \begin{array}{rcl} y & = & m x + b \\ y & = & 3 x + 2 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} y &=& {\color{blue}m}x +{\color{red}b} \\ y &=& {\color{blue}3}x+{\color{red}2} \end{array}\end{align*}$
::y=mx+by=3x+2 y=mx+by=3x+2

The slope of the line is ‘3’ and the $\begin{align*}y\end{align*}$ -intercept is 2.
::线的斜坡是“3”,y-interview是“2”。

Example 3
::例3

For the following equation written in slope-intercept form, what is the slope and the $\begin{align*}y\end{align*}$ -intercept of the line?
::对于以斜坡间隙形式写成的以下方程式,斜坡和横线的 Y 间隙是什么?

$\begin{aligned} y = - \frac{3}{4} x - 1 \end{aligned}$
$\begin{align*}y=-\frac{3}{4} x-1\end{align*}$
::y34x-1

First, match the given equation with the general form of an equation written in slope-intercept form.
::首先,将给定方程式与以斜坡界面写成的方程式的一般形式相匹配。

$\begin{aligned} \begin{array}{rcl} y & = & m x + b \\ y & = & - \frac{3}{4} x - 1 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} y &=& {\color{blue}m}x +{\color{red}b} \\ y &=& -{\color{blue}\frac{3}{4}}x - {\color{red}1} \end{array}\end{align*}$
::y=mx+by=34x-1

The slope of the line is represented by ‘ $\begin{align*}m\end{align*}$ ’ and has a value of $\begin{align*}{\color{blue}-}{\color{blue}\frac{3}{4}}\end{align*}$ . The $\begin{align*}y\end{align*}$ -intercept of the line is represented by ‘ $\begin{align*}b\end{align*}$ ’ and has a value of $\begin{align*}{\color{red}-1}\end{align*}$ .
::线的斜坡以`m ' 表示,其值为-34。线的Y插口以`b ' 表示,其值为-1。

Example 4
::例4

Write the following equation in slope-intercept form and state the slope and the $\begin{align*}y\end{align*}$ -intercept of the line.
::以斜坡截取形式写下以下方程式,并标明斜坡和横线的 Y 截取。

$\begin{aligned} 7 x - 3 y = 21 \end{aligned}$
$\begin{align*}7x - 3y = 21\end{align*}$
::7-3y=21

First, subtract $\begin{align*}7x\end{align*}$ from both sides of the equation and simplify.
::首先,从方程两侧减去7x并简化。

$\begin{aligned} \begin{array}{rcl} 7 x - 3 y & = & 21 \\ 7 x - 7 x - 3 y & = & 21 - 7 x \\ - 3 y & = & 21 - 7 x \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 7x - 3y &=& 21 \\ 7x - 7x - 3y &=& 21-7x \\ -3y &=& 21 - 7x \end{array}\end{align*}$
::7-3y=217x-7x-3y=21-7x-3y=21-7x-7x-3yy=21-7x

Next, divide both sides of the equation by ‘-3’ to solve for ‘ $\begin{align*}y\end{align*}$ ’.
::接下来,将方程式两边除以 ' 3 ' ,

$\begin{aligned} \begin{array}{rcl} - 3 y & = & 21 - 7 x \\ \frac{\overset{1}{- 3} y}{- 3} & = & \frac{\overset{- 7}{21}}{- 3} - \frac{7}{- 3} x \\ y & = & - 7 + \frac{7}{3} x \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} -3y &=& 21 - 7x \\ \frac{\overset{1} {\cancel{-3}}y}{{\cancel{-3}}} &=& \frac{\overset{-7} {\cancel{21}}}{{\cancel{-3}}} - \frac{7}{-3} x \\ y &=& -7 + \frac{7}{3} x \end{array}\end{align*}$
::-3y=21-7x-31y-3=21-7-3-7-3-7-3-3xy=7+73x

Then, express the equation in slope-intercept form.
::然后用斜坡界面表达方程式

$\begin{aligned} \begin{array}{rcl} y & = & m x + b \\ y & = & \frac{7}{3} x - 7 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} y &=& {\color{blue}m}x +{\color{red}b} \\ y &=& {\color{blue}\frac{7}{3}} x -{\color{red}7} \end{array}\end{align*}$
::y=mx+by=73x-7 y=mx+by=73x-7

The slope of the line is $\begin{align*}\frac{7}{3}\end{align*}$ and the $\begin{align*}y\end{align*}$ -intercept is -7.
::线的斜坡是73, y 接口是 -7。

Example 5
::例5

Write the following equation in slope-intercept form and state the slope and the $\begin{align*}y\end{align*}$ -intercept of the line.
::以斜坡截取形式写下以下方程式,并标明斜坡和横线的 Y 截取。

$\begin{aligned} 5 x + y = - 10 \end{aligned}$
$\begin{align*}5x + y = -10\end{align*}$
::5x+y10

First, subtract $\begin{align*}5x\end{align*}$ from both sides of the equation and simplify.
::首先,从方程两侧减去5x并简化。

$\begin{aligned} \begin{array}{rcl} 5 x + y & = & - 10 \\ 5 x - 5 x + y & = & - 10 - 5 x \\ y & = & - 10 - 5 x \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 5x + y &=& -10 \\ 5x - 5x + y &=& -10 -5x \\ y &=& -10 -5x \end{array}\end{align*}$
::5x+y 105x-5x+y 10-5xy 10-5xy 10-5xx

Then, express the equation in slope-intercept form.
::然后用斜坡界面表达方程式

$\begin{aligned} \begin{array}{rcl} y & = & m x + b \\ y & = & - 5 x - 10 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} y &=& {\color{blue}m}x +{\color{red}b} \\ y &=& {\color{blue}-5}x -{\color{red}10} \end{array}\end{align*}$
::y=mx+by=%5x-10 y=mx+by=%5x-10

The slope of the line is -5 and the $\begin{align*}y\end{align*}$ -intercept is -10.
::线的斜坡是 -5 y 拦截是 -10

Review
::回顾

Look at each equation and identify the slope and the $\begin{align*}y\end{align*}$ -intercept by looking at each equation. There are two answers for each problem.
::查看每个方程式, 并用查看每个方程式来识别斜度和 Y 界面。每个问题有两个答案。

1. $\begin{align*}y = 2x + 4\end{align*}$
::1. y=2x+4

2. $\begin{align*}y = 3x-2\end{align*}$
::2. y=3x-2

3. $\begin{align*}y = 4x + 3\end{align*}$
::3.y=4x+3 y=4x+3

4. $\begin{align*}y = 5x-1\end{align*}$
::4.y=5x-1

5. $\begin{align*}y = \frac{1}{2} x + 2\end{align*}$
::5.y=12x+2

6. $\begin{align*}y = -2x + 4\end{align*}$
::6. y2x+4

7. $\begin{align*}y = -3x-1\end{align*}$
::7. y3x-1

8. $\begin{align*} y = \frac{-1}{3} x + 5\end{align*}$
::8. y13x+5

Use what you have learned to write each in slope-intercept form and then answer each question.
::使用您所学的知识,以斜坡界面形式写出每个问题,然后回答每个问题。

9. What form is the following equation in: $\begin{align*}2x+4y=12\end{align*}$
::9. 2x+4y=12的下列方程式形式如下:

10. Write this equation in slope-intercept form.
::10. 以斜坡界面形式写出这一方程。

11. What is the slope?
::11. 什么是斜坡?

12. What is the $\begin{align*}y\end{align*}$ -intercept?
::12. 什么是Y - 接口?

13. What form is the following equation in: $\begin{align*}6x+3y=24\end{align*}$
::13. 6x+3y=24的下列方程式形式如下:

14. Write this equation in slope-intercept form.
::14. 以斜坡界面形式写出这个方程。

15. What is the slope?
::15. 什么是斜坡?

16. What is the $\begin{align*}y \end{align*}$ -intercept?
::16 什么是Y - 界面?

17. What form is the following equation in: $\begin{align*}5x+5y=15\end{align*}$
::17. 5x+5y=15

18. Write this equation in slope-intercept form.
::18. 以斜坡拦截形式写下这个方程。

19. What is the slope?
::19. 斜坡是什么?

20. What is the $\begin{align*}y\end{align*}$ -intercept?
::20. 什么是Y的接口?

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

使用斜坡截取表

Slope Intercept Form ::斜坡截截表

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Slope Intercept Form
::斜坡截截表

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)