Section: 标准表格 | 2.3 标准表格

Section outline

At a football game, Brian is selling concessions. He sells soda for $2 apiece and popcorn for $3 per bag. At the end of the night, he has sold out of soda and has made a total of $264. If he sold 60 sodas, how many bags of popcorn did he sell?
::在一场足球赛上,Brian卖的是特许权。他卖的是苏打汽水,每块2美元,爆米花每袋3美元。到了晚上,他卖完了苏打水,总共赚了264美元。如果他卖了60苏打水,他卖了多少袋爆米花?

Standard Form
::标准表格

Slope-intercept form is one way to write the equation of a line. Another way is called standard form . Standard form looks like $\begin{align*}Ax + By = C,\end{align*}$ where $\begin{align*}A, B,\end{align*}$ and $\begin{align*}C\end{align*}$ are all real numbers.
::斜截面表是写行方程式的一种方式。另一种方式称为标准格式。标准格式看起来像 Ax+By=C, A、 B和 C 都是真实数字。

Let's f ind the equation of a line, in standard form, where the is $\begin{align*}\frac{3}{4}\end{align*}$ and passes through (4, -1).
::让我们找到一条线的方程, 以标准格式, 以34为标准格式, 然后通过( 4, - 1) 。

To find the equation in standard form, you need to determine what $\begin{align*}A, B,\end{align*}$ and $\begin{align*}C\end{align*}$ are. Let’s start this problem by finding the equation in slope-intercept form.
::要找到标准形式的方程, 您需要确定 A、 B 和 C 是什么。让我们从以斜度界面查找方程开始解决问题。

$\begin{aligned} - 1 & = \frac{3}{4} (4) + b \\ - 1 & = 3 + b \\ - 4 & = b \end{aligned}$
$\begin{align*}-1 &= \frac{3}{4}(4) + b\\ -1 &= 3 + b \\ -4 &= b\end{align*}$
::-1=34(4)+b-1=3+b-4=b

In slope-intercept form, the equation is $\begin{align*}y = \frac{3}{4}x-4\end{align*}$ .
::在斜坡界面形式中,公式为y=34x-4。

To change this to standard form we need to subtract the $\begin{align*}x-\end{align*}$ term from both sides of the equation.
::要将此修改为标准格式,我们需要从等式的两侧减去x-条件。

$\begin{aligned} - \frac{3}{4} x + y = - 4 \end{aligned}$
$\begin{align*}- \frac{3}{4}x+y = -4\end{align*}$
::- 34x+y4

Let's find the slope and $\begin{align*}y-\end{align*}$ intercept if t he equation of a line is $\begin{align*}5x-2y = 12\end{align*}$ .
::如果一条线的方程式是 5x--2y=12,让我们找到斜坡和 y- interview。

To find the slope and $\begin{align*}y-\end{align*}$ intercept of a line in standard form, we need to switch it to slope-intercept form. This means, we need to solve the equation for $\begin{align*}y\end{align*}$ .
::为了在标准格式中找到一条线的斜度和 y- interview, 我们需要将其切换为斜度界面。这意味着, 我们需要解析 y 的方程。

$\begin{aligned} 5 x - 2 y & = 12 \\ - 2 y & = - 5 x + 12 \\ y & = \frac{5}{2} x - 6 \end{aligned}$
$\begin{align*}5x - 2y &= 12\\ -2y &= -5x + 12 \\ y &= \frac{5}{2}x - 6\end{align*}$
::5x-2y=12-2y5x+12y=52x-6

From this, the slope is $\begin{align*}\frac{5}{2}\end{align*}$ and the $\begin{align*}y-\end{align*}$ intercept is (0, -6).
::从这里起,斜坡为52, y - 拦截为( 0, 6) 。

Now, let's find the equation of the line, in standard form, using the graph below.
::现在,让我们用下面的图表, 以标准的形式, 找到线的方程式。

Here, we are given the intercepts . The slope triangle is drawn by the axes, $\begin{align*}\frac{-6}{-2} = 3\end{align*}$ . And, the $\begin{align*}y-\end{align*}$ intercept is (0, 6). The equation of the line, in slope-intercept form, is $\begin{align*}y = 3x + 6\end{align*}$ . To change the equation to standard form, subtract the $\begin{align*}x-\end{align*}$ term to move it over to the other side.
::在这里,我们得到了拦截。斜度三角形是由轴绘制的, - 6 - 2= 3; y - 截取是 0, 6 。斜度截取形式的线的方程式是 y= 3x+6。要将方程式转换为标准格式, 请减去 x - term, 将其移到另一侧。

$\begin{aligned} - 3 x + y = 6 o r 3 x - y = - 6 \end{aligned}$
$\begin{align*}-3x + y = 6 \ or \ 3x - y = -6\end{align*}$
::- 3x+y=6或3x-y6

Let's find the intercepts if the equation of a line is $\begin{align*}6x - 5y = 45\end{align*}$ .
::如果一条线的方程式是 6x -5y=45,我们去找拦截器

For the $\begin{align*}x-\end{align*}$ intercept, the $\begin{align*}y-\end{align*}$ value is zero. Plug in zero for $\begin{align*}y\end{align*}$ and solve for $\begin{align*}x\end{align*}$ .
::x- interfict 的 Y - 值为 0。 y 的插件为 0 , x 的解析符为 X 。

$\begin{aligned} 6 x - 5 y & = 45 \\ 6 x - 5 (0) & = 45 \\ 6 x & = 45 \\ x & = \frac{45}{6} o r \frac{15}{2} \end{aligned}$
$\begin{align*}6x - 5y &= 45\\ 6x - 5(0) &= 45\\ 6x &= 45 \\ x &= \frac{45}{6} \ or \ \frac{15}{2}\end{align*}$
::6x-5y=456x-5(0)=456x=455x=456x或152

The $\begin{align*}x-\end{align*}$ intercept is $\begin{align*}\left(\frac{15}{2}, 0 \right)\end{align*}$ .
::x - 拦截( 152, 0) 。

For the $\begin{align*}y-\end{align*}$ intercept, the $\begin{align*}x-\end{align*}$ value is zero. Plug in zero for $\begin{align*}x\end{align*}$ and solve for $\begin{align*}y\end{align*}$ .
::对于 y - intercut, x - 值为 0。 x 的插件为 0 , y 的解答为 y 。

$\begin{aligned} 6 x - 5 y & = 45 \\ 6 (0) - 5 y & = 45 \\ - 5 y & = 45 \\ y & = - 9 \end{aligned}$
$\begin{align*}6x -5y &= 45\\ 6(0) - 5y &= 45\\ -5y &= 45 \\ y &= -9\end{align*}$
::6x-5y=456(0)-5y=45-5y=45y=45y=45y9

The $\begin{align*}y-\end{align*}$ intercept is (0, -9).
::y- 拦截是 0, - 9 。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the number of bags of popcorn Brian sold.
::之前有人要你找到布赖恩卖的爆米花包数

This type of problem is easier to write in standard form. We will call x the number of sodas and y the number of bags of popcorn. The coefficients are the cost of each item in dollars. The expression will equal the total amount of concessions Brian sold.
::这种类型的问题比较容易以标准格式书写。我们将调用 x 苏打水的数量和 y 袋爆米花的数量。系数是每个物品的成本( 美元) 。表达方式将等于 Brian 出售的特许权总额。

$\begin{align*}2x+3y=264\end{align*}$
::2x+3y=264

Now, we know that $\begin{align*}x=60\end{align*}$ . Plug that in and solve for y.
::现在,我们知道x=60。插进去,解决你的问题。

$\begin{aligned} 2 (60) + 3 y & = 264 \\ 120 + 3 y & = 264 \\ 3 y & = 144 \\ y & = 48 \end{aligned}$
$\begin{align*}2(60)+3y&=264 \\ 120+3y&=264\\ 3y&=144\\ y&=48\end{align*}$
::2(60)+3y=264120+3y=2643y=144y=48

Therefore , Brian sold 48 bags of popcorn.
::因此,布莱恩卖了48袋爆米花

Example 2
::例2

Find the equation of the line, in standard form, that passes through (8, -1) and (-4, 2).
::以标准格式查找通过(8,-1)和(4,2)的直线方程式。

We need to first find the equation of this line in $\begin{align*}y-\end{align*}$ intercept form and then change it to standard form. First, find the slope.
::我们需要首先在y- intermission 中找到这条线的方程式, 然后将其修改为标准格式。首先, 找到斜坡。

$\begin{aligned} \frac{2 - (- 1)}{- 4 - 8} = \frac{3}{- 12} = - \frac{1}{4} \end{aligned}$
$\begin{align*}\frac{2-(-1)}{-4-8} = \frac{3}{-12} = - \frac{1}{4}\end{align*}$

Find the $\begin{align*}y-\end{align*}$ intercept using slope-intercept form.
::使用斜坡截取形式查找 y- interphysict 。

$\begin{aligned} 2 & = - \frac{1}{4} (- 4) + b \\ 2 & = 1 + b \\ 1 & = b \end{aligned}$
$\begin{align*}2 &= - \frac{1}{4}(-4) + b\\ 2 &= 1 + b \\ 1 &= b\end{align*}$
::214(-4)+b2=1+b1=b

The equation of the line is $\begin{align*}y = - \frac{1}{4}x + 1\end{align*}$ .
::线的方程式是 y14x+1 。

To change this equation into standard form, add the $\begin{align*}x-\end{align*}$ term to both sides.
::为了将这一等式改变为标准形式,在双方增加x-条件。

$\begin{aligned} \frac{1}{4} x + y = 1 \end{aligned}$
$\begin{align*}& \quad \frac{1}{4}x + y = 1\end{align*}$
::14x+y=1 14x+y=1

Example 3
::例3

Change $\begin{align*}2x + 3y = 9\end{align*}$ to slope-intercept form.
::将 2x+3y=9 更改为斜坡界面。

To change $\begin{align*}2x + 3y = 9\end{align*}$ into slope-intercept form, solve for $\begin{align*}y\end{align*}$ .
::将 2x+3y=9 修改为斜坡界面,为 y 求解。

$\begin{aligned} 2 x + 3 y & = 9 \\ 3 y & = - 2 x + 9 \\ y & = - \frac{2}{3} x + 3 \end{aligned}$
$\begin{align*}2x + 3y &=9\\ 3y &= -2x + 9\\ y &= - \frac{2}{3}x + 3\end{align*}$
::2x+3y=93y2x+9y23x+3

Example 4
::例4

What are the intercepts of $\begin{align*}3x - 4y = -24\end{align*}$ ?
::3x -4y24的拦截是什么?

First, plug in zero for $\begin{align*}y\end{align*}$ and solve for $\begin{align*}x\end{align*}$ .
::首先,插入 y 零, 解答 x 。

$\begin{aligned} 3 x - 4 (0) & = - 24 \\ 3 x & = - 24 \\ x & = - 8 \end{aligned}$
$\begin{align*}3x - 4(0) &= -24\\ 3x &= -24 \\ x &= -8\end{align*}$
::3x-4(0)24x8

$\begin{align*}x-\end{align*}$ intercept is (-8, 0)
::x- 拦截是 (-8, 0)

Now, start over and plug in zero for $\begin{align*}x\end{align*}$ and solve for $\begin{align*}y\end{align*}$ .
::现在,从头开始,在零插入 x 和为y 解决问题。

$\begin{aligned} 3 (0) - 4 y & = - 24 \\ - 4 y & = - 24 \\ y & = 6 \end{aligned}$
$\begin{align*}3(0) - 4y &=-24\\ -4y &= -24 \\ y &= 6\end{align*}$
::3(0)-4y24-4y24y=6

$\begin{align*}y-\end{align*}$ intercept is (6, 0)
::y- 拦截( 6, 0)

Review
::回顾

Change the following equations into standard form.
::将以下方程式改变为标准形式。

$\begin{align*}y = - \frac{2}{3}x + 4\end{align*}$
::y 23x+4

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

$\begin{align*}y = \frac{1}{5}x - 1\end{align*}$
::y=15x-1

Change the following equations into slope-intercept form.
::将以下方程式改变为斜坡界面形式。

$\begin{align*}4x + 5y = 20\end{align*}$
::4x+5y=20

$\begin{align*}x - 2y = 9\end{align*}$
::x-2y=9

$\begin{align*}2x -3y = 15\end{align*}$
::2x-3y=15

Find the $\begin{align*}x\end{align*}$ and $\begin{align*}y-\end{align*}$ intercepts of the following equations.
::查找以下方程式的 x 和 y - 界面。

$\begin{align*}3x + 4y = 12\end{align*}$
::3x+4y=12

$\begin{align*}6x - y = 8\end{align*}$
::6-y=8

$\begin{align*}3x + 8y = -16\end{align*}$
::3x+8y16

Find the equation of the lines below, in standard form.
::以标准格式查找以下行的方程式。

slope = 2 and passes through (3, -5)
::斜坡=2,通过(3,5)

slope $\begin{align*}= - \frac{1}{2}\end{align*}$ and passes through (6, -3).
::斜坡 12 并通过 (6, 3) 。

passes through (5, -7) and (-1, 2)
::通过(5,-7)和(-1,2)

passes through (-5, -5) and (5, -3)
::通过(5-5,5-5)和(5,3)

Change $\begin{align*}Ax + By = C\end{align*}$ into slope-intercept form.
::将 Ax+By=C 更改为斜坡界面。

From #16, what are the slope and $\begin{align*}y-\end{align*}$ intercept equal to (in terms of $\begin{align*}A, B,\end{align*}$ and/or $\begin{align*}C\end{align*}$ )?
::从16号公路起,斜坡和y - inter- interfiles 等于(A、B和/或C)什么?

Using #16 and #17, find one possible combination of $\begin{align*}A, B,\end{align*}$ and $\begin{align*}C\end{align*}$ for $\begin{align*}y = \frac{1}{2}x - 4\end{align*}$ . Write your answer in standard form.
::使用 #16 和 # 17, 找到一个可能的 A、 B 和 C 组合, 用于 y= 12x- 4。请以标准格式填写您的回答。

The measure of a road’s slope is called the grade . The grade of a road is measured in a percentage, for how many vertical feet the road rises or declines over 100 feet. For example, a road with a grade incline of 5% means that for every 100 horizontal feet the road rises 5 vertical feet. What is the slope of a road with a grade decline of 8%?
::道路坡度的测量称为“等级 ” 。道路的等级以百分比衡量,因为道路的垂直坡度上升或下降超过100英尺。比如,一个等级为5 % 的公路意味着每100个水平坡度,道路就上升5英尺。等级下降8 % 的公路坡度是多少?

The population of a small town in northern California gradually increases by about 50 people a year. In 2010, the population was 8500 people. Write an equation for the population of this city and find its estimated population in 2017.
::北加利福尼亚州一个小城镇的人口每年逐步增加约50人,2010年人口为8,500人,为该城市的人口写一个方程式,并在2017年找到其估计人口。

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

Standard Form ::标准表格

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Standard Form
::标准表格

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)