Course: 3.7 书写和溶解线性线性赤道

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书写和溶解线性线性公式
Equations All Around Us
::我们周围的赤道

In , you have seen many formulas and equations that are used in business. Some of these equations are commonly known, but some have to be created to go in spreadsheets and computer programs. Outside of the business world, many professions require the ability to use equations to represent real-world situations. For example, all scientists need to be able to use math to represent the mathematical relationships of the events in their respective fields. The movement of every object around you can be expressed using an equation . If that seems hard to believe, think of a video game. Video game designers use math equations to simulate the entire world around us. Equations determine every movement and object in a video game. In this section, you will be using equations to represent and answer questions about mathematical relationships.
::在商务中,您已经看到了许多用于商业的公式和方程式。有些这些方程式是众所周知的,但有些方程式必须创建成电子表格和计算机程序。在商业界以外,许多专业需要能够使用方程式来代表现实世界的情况。例如,所有科学家都需要能够使用数学来代表其各自领域的事件的数学关系。每个对象在您周围的移动都可以使用方程式来表达。如果这看起来很难相信,可以想象一个电子游戏。视频游戏设计者使用数学方程式来模拟我们周围的整个世界。等式决定了游戏中的每一个运动和对象。在这个节中,你将使用方程式来表达和回答关于数学关系的问题。

Writing and Solving Linear Equations
::书写和溶解线性线性公式

T o develop the skills necessary to write linear equations, you will look at examples using many contexts. To begin, explore the relationships between numbers and expressions in their most basic form.
::要开发写入线性方程式所需的技能, 您将使用多种背景来查看示例。首先, 探索数字和表达形式之间以最基本的形式存在的关系。

Example
::示例示例示例示例

Three times what number is equal to 8 less than seven times the number?
::数字是数字的三倍,是数字的八倍,比数字的七倍少七倍?

Begin by breaking this down piece by piece.
::开始逐个打破这个碎片。

$\begin{aligned} ∙ \end{aligned}$ Three times what number: $\begin{aligned} 3 x \end{aligned}$
::* 数目的三倍:3x

$\begin{aligned} ∙ \end{aligned}$ is equal to: $\begin{aligned} = \end{aligned}$
::等于:==

$\begin{aligned} ∙ \end{aligned}$ eight less than: $\begin{aligned} - 8 \end{aligned}$ (subtrac t 8)
::少于:-8(减号8)

$\begin{aligned} ∙ \end{aligned}$ seven times the number: $\begin{aligned} 7 x \end{aligned}$
::* 数字的七倍:7倍

Putting it all together you get the following equation:
::把它们拼凑在一起,你得到的公式如下:

$\begin{aligned} 3 x = 7 x - 8 \end{aligned}$

::3x=7x-8

You can solve this using the skills we have learned in previous sections.
::您可以使用我们在前几节学到的技能来解决这个问题。

$\begin{aligned} 3 x & = 7 x - 8 \\ - 4 x & = - 8 \\ x & = 2 \end{aligned}$

::3x=7x-8-4x=2

Rubber Ducky Math
::橡胶鸭

In this PLIX interactive, you will use a tape diagram to help you solve an equation with variables on both sides.
::在此 PLIX 互动中, 您将使用磁带图来帮助您解析一个包含两个侧变量的方程式。

Geometry Problems
::几何问题

T hese problems often pop up in geometric contexts in professions like architecture, engineering, carpentry, interior decorating and more. In situations like this, the best approach is to find a formula that fits the situation and substitute in the values or expressions for each variable .
::这些问题往往出现在建筑、工程、木工、内部装饰等行业的几何环境中。在这样的情况下,最好的办法是找到一个适合形势的公式,并取代每个变量的数值或表达方式。

Example
::示例示例示例示例

A rectangular yard has 448 feet of fencing around the perimeter. The width is five more than double the length. What are the dimensions of the property ?
::长方形院子周围有448英尺的围栏,宽度为5倍以上。地产的尺寸是多少?

Since you are dealing with the perimeter of a rectangle, use the following equation:
::既然你正在处理一个矩形的周边,请使用以下方程式:

$\begin{aligned} Perimeter = 2(Length + Width) \end{aligned}$

::周边 = 2( 林特 + 宽度)

First, you should state what you know. You know right away that the perimeter is 448 feet because you are directly told this in the problem. However, what does “width is five more than double the length” mean? B reak it down as you did above.
::首先,你应该说明你所知道的。你马上知道周界是448英尺,因为这个问题直接告诉了你。然而,“宽度是5倍以上两倍”意味着什么?按上面的写法把它拆开。

$\begin{aligned} ∙ \end{aligned}$ Width: $\begin{aligned} W \end{aligned}$
::宽度: W

$\begin{aligned} ∙ \end{aligned}$ is: =
::系指:==

$\begin{aligned} ∙ \end{aligned}$ five more than: $\begin{aligned} + 5 \end{aligned}$ (add 5)
::* 5个以上:+5(加5)

$\begin{aligned} ∙ \end{aligned}$ double the length: $\begin{aligned} 2 L \end{aligned}$
::长度加倍:2升

When you put these together, you will see the following equation:
::当你把这些组合起来时,你会看到以下的方程:

$\begin{aligned} W = 2 L + 5 \end{aligned}$

::W=2L+5

The purpose of this equation is to tell you what $\begin{aligned} W \end{aligned}$ is in terms of $\begin{aligned} L \end{aligned}$ . You can substitute this into the equation $\begin{aligned} Perimeter = 2(Length + Width) . \end{aligned}$
::这个方程的目的是要告诉您在L.方面W是什么。您可以将它替换为 Peritom = 2 (Length + Width) 的方程。

$\begin{aligned} ∙ \end{aligned}$ Perimeter: 448
::* 周边:448

$\begin{aligned} ∙ \end{aligned}$ Length: $\begin{aligned} L \end{aligned}$
::* 长度:L

$\begin{aligned} ∙ \end{aligned}$ Width: $\begin{aligned} 2 L + 5, \end{aligned}$ write what $\begin{aligned} W \end{aligned}$ is equivalent to in terms of $\begin{aligned} L \end{aligned}$
::* Width: 2L+5, 写下按L值计算W相当于W值的文字

When you substitute these into the formula $\begin{aligned} Perimeter = 2(Length + Width), \end{aligned}$ you get the following:
::当您将这些替换为公式周围值 = 2( Length + Width) = 2( length + Width)时,您可以得到以下信息:

$\begin{aligned} 448 = 2 (L + 2 L + 5) \end{aligned}$

::448=2(L+2L+5)

S olve this equation using what you have learned in . In this case, since there are two like terms inside the parenthesis, you can combine before you distribute.
::使用您所学的知识来解析这个方程式。在此情况下, 由于括号中有两个类似条件, 您可以在分配前合并。

$\begin{aligned} 448 & = 2 (3 L + 5) \\ 448 & = 6 L + 10 \\ 438 & = 6 L \\ 73 & = L \end{aligned}$

::448=2(3L+5-5)448=6L+10438=6L73=L

The length is 73 feet, but you do not know the width. You can use the length to find the width since you know that width is five more than double the length.
::长度为73英尺,但您不知道宽度。您可以使用长度找到宽度,因为您知道宽度是长度的五倍以上。

$\begin{aligned} W = 2 (73) + 5 = 151 \end{aligned}$

::W=2(273+5=151)

The length is 73 feet and the width is 151 feet.
::长度73英尺宽度151英尺

Pick a Truck
::选一辆卡车

Zoe De Leon owns a chain of grocery stores located in the Northeast United States. She needs to buy tractor trailer trucks to deliver the food from the warehouse to the stores. She is trying to choose between two different trucks. The first truck costs $145,000 and will cost $0.31 per mile traveled base on the current gas prices. The second truck will cost $160,000 but gets better gas mileage, costing $0.27 per mile. Which truck should she buy?
::Zoe De Leon拥有位于美国东北部的一连串杂货店。她需要购买拖拉机拖车,将食品从仓库运到仓库。她试图在两辆不同的卡车之间做出选择。第一辆卡车需要145 000美元,按目前天然气价格每英里旅行0.31美元。第二辆卡车需要160 000美元,但得到更好的天然气里程,每英里2.27美元。她应该买哪辆卡车?

U se an equation to find the number of miles ( $\begin{aligned} m \end{aligned}$ ) driven, such that the cost of the trucks will be equal. What equation can you write to help determine when the total cost of the trucks is equal?
::使用方程来查找驱动的英里数, 这样卡车的成本就会相等。您能写出什么方程来帮助确定卡车的总成本是相等的?

You will first need to find the total cost of each truck.
::你首先需要找出每辆卡车的总成本。

$\begin{aligned} ∙ \end{aligned}$ Truck 1 costs $145,000 and $0.31 per mile : $\begin{aligned} 145000 + 0.31 m \end{aligned}$
::* 1卡车每英里145,000美元和0.31美元:145000+0.31米

$\begin{aligned} ∙ \end{aligned}$ Truck 2 costs $160,000 and $0.27 per mile : $\begin{aligned} 160000 + 0.27 m \end{aligned}$
::* 2卡车费用160 000美元和每英里0.27美元:160 000+0.27米

To find when these values will be equal you can simply set the expressions equal to each other giving you the following equation:
::要找到这些值何时会相等, 您可以简单地设置对等的表达式, 给您以下等式 :

$\begin{aligned} 145000 + 0.31 m = 160000 + 0.27 m \end{aligned}$

::145000+0.31m=160000+0.27m

You can then solve to obtain how many miles the trucks will have to drive for the costs of the trucks to be equal.
::然后,你就可以解决,以获得卡车要开多少英里才能支付卡车的同等费用。

$\begin{aligned} 145000 + 0.31 x & = 160000 + 0.27 x \\ 145000 + 0.04 x & = 160000 \\ 0.04 x & = 15000 \\ x & = 375000 \end{aligned}$

::145000+0.31x=160000+0.27x145000+0.04x=1600000.004x=1500x=375000

If each truck drives 375,000 miles, the two trucks will cost the same amount.
::如果每辆卡车开375 000英里,两辆卡车的费用也将相同。

Discussion Questions
::讨论问题讨论问题

Which truck should Zoe buy? In which situations would one truck be better than the other?
::Zoe应该买哪辆卡车?

Why would Zoe buy the $145000 truck even though it is the more expensive long term ?
::为何佐伊要买这辆$145000的卡车?

Consecutive Integer Problems
::连续整数问题

Consecutive integer problems aren’t as common in most professions but the skills needed to solve them are. The most important skill involved in consecutive integer problems is the ability to write multiple variables in terms of one variable. This allows you to solve the equation for one of the variables and use the relationships to figure out the remaining variables.
::连续整数问题在多数职业中并不常见,但解决它们所需要的技能却是。连续整数问题中最重要的技能是能够用一个变量来写出多个变量。这让您能够解答其中一个变量的方程,并利用关系来找出剩余变量。

Example
::示例示例示例示例

Find three consecutive integers with a sum of 51.
::查找三个连续整数,总和为51。

All you know is that the numbers are consecutive integers and they add to 51. If you know that the first number, or least number, is 7, you can figure out that the other two numbers are 8 and 9. No tice how the second number is one more than the first number, and the third number is two more than the first number. If you assume that the first number is $\begin{aligned} x, \end{aligned}$ you can write expressions for the next numbers in terms of $\begin{aligned} x . \end{aligned}$
::您只知道数字是连续的整数,它们会添加到51。如果您知道第一个数字或最小的数字是7,您可以发现其他两个数字是8和9。请注意第二个数字是1比第一个数字多,第三个数字是2比第一个数字多。如果您假设第一个数字是x,您可以用 x 来写下一个数字的表达式。

$\begin{aligned} ∙ \end{aligned}$ First Number: $\begin{aligned} x \end{aligned}$
::* 第一编号:x

$\begin{aligned} ∙ \end{aligned}$ Second Number: $\begin{aligned} x + 1 \end{aligned}$
::* 第二号号码:x+1

$\begin{aligned} ∙ \end{aligned}$ Third Number: $\begin{aligned} x + 2 \end{aligned}$
::* 第三号号码:x+2

Now that you have written an expression for each number in terms of one variable, you can write an equation for the sum of these three numbers equaling 51.
::既然您已经用一个变量为每个数字写了一个表达式, 您可以为这三个数字的和写一个等式, 共等于51。

$\begin{aligned} x + x + 1 + x + 2 = 51 \end{aligned}$

::x+x+1+x+2=51

You can then solve for $\begin{aligned} x . \end{aligned}$
::然后您可以解答 x 。

$\begin{aligned} x + x + 1 + x + 2 & = 51 \\ 3 x + 1 + 2 & = 51 \\ 3 x + 3 & = 51 \\ 3 x & = 48 \\ x & = 16 \end{aligned}$

::x+x+1+x+2=513x+1+2=513x+3=513x=513x=513x=48x=16

Knowing that the first number is 16, you can figure out that the next two numbers are 17 and 18.
::知道第一个数字是16, 你可以发现接下来的两个数字是17和18。

Answer: 16, 17, 18
::答复:16、17、18

These problems can also involve relationships between odd and even numbers.
::这些问题还可能涉及奇数和偶数之间的关系。

Consecutive Integers: $\begin{aligned} x, x + 1, x + 2, . . . \end{aligned}$
::后继整数: x,x+1,x+2,...

Consecutive Even Integers (where $\begin{aligned} x \end{aligned}$ is an even number): $\begin{aligned} x, x + 2, x + 4, . . . \end{aligned}$
::连整数( x 是偶数) x,x+2,x+4,...

Consecutive Odd Integers (where $\begin{aligned} x \end{aligned}$ is an odd number): $\begin{aligned} x, x + 2, x + 4, . . . \end{aligned}$
::连续的奇数整数( x 是奇数) : x, x+2, x+4,...

Discussion Question
::讨论问题

Why aren’t the expressions for consecutive odd integers $\begin{aligned} x, x + 1, x + 3, . . . ? \end{aligned}$
::为什么连续奇数整数 x, x+1, x+3 的表达式不是...?

Summary
::摘要

Equations with variables can be used to represent real-world situations.
::与变量的等式可以用来代表现实世界的情况。

Section outline

General

书写和溶解线性线性公式

Equations All Around Us ::我们周围的赤道

Writing and Solving Linear Equations ::书写和溶解线性线性公式

Rubber Ducky Math ::橡胶鸭

Geometry Problems ::几何问题

Pick a Truck ::选一辆卡车

Consecutive Integer Problems ::连续整数问题

Summary ::摘要

Equations All Around Us
::我们周围的赤道

Writing and Solving Linear Equations
::书写和溶解线性线性公式

Rubber Ducky Math
::橡胶鸭

Geometry Problems
::几何问题

Pick a Truck
::选一辆卡车

Consecutive Integer Problems
::连续整数问题

Summary
::摘要