课程： 1.1 变量的定义

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选择节变量变量的定义

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变量变量的定义
Imagine you have a jar filled with dimes and quarters, and know that the total of the 56 coins in the jar is $8.60. It is possible to determine how many of each coin the jar contains by using Algebra if you know how to represent the situation mathematically.

Variables
::变量变量变量

No one likes doing the same problem over and over again, that’s why mathematicians invented algebra. Algebra takes the basic principles of math and makes them more general, so a problem can be solved once and then that solution used to solve a group of similar problems.
::没有人喜欢一次又一次地做同样的问题,这就是数学家发明代数的原因。代数采用数学的基本原则,使其更加普遍,这样问题就可以一而再而三地解决,解决一系列类似的问题。

In arithmetic, you’ve dealt with numbers and their arithmetical operations (such as $\begin{aligned} +, -, \times, \div \end{aligned}$ ). In algebra, we often use symbols called variables (which are usually letters, such as $\begin{aligned} x, y, a, b, c, \dots \end{aligned}$ ) to represent numbers, and sometimes processes.
::在算术中,您处理的是数字及其算术操作(例如 +, -, ~, ~, ~, ~) 。在代数中,我们经常使用称为变量的符号(通常是字母,例如 x, y, a, b, c,...)来表示数字,有时是过程。

For example, we might use the letter $\begin{aligned} x \end{aligned}$ to represent some number we don’t know yet, but need to identify in the course of a problem. Or we might use two letters, like $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y, \end{aligned}$ to show a relationship between two numbers without needing to know what the actual numbers are. The same letters can represent a wide range of possible numbers, and the same letter may represent completely different numbers when used in two different problems.
::例如,我们也许可以用字母x来代表一些我们尚不知道的数字,但在问题过程中需要识别。或者我们可以用两个字母,例如x和y,来显示两个数字之间的关系,而不必知道实际数字是什么。同一个字母可以代表广泛的可能数字,而同一字母在两个不同的问题中使用时可能代表完全不同的数字。

Using variables offers advantages over solving each problem “from scratch.” With variables, we can:
::使用变量为“从零开始”解决每个问题提供了优势。有了变量,我们可以:

Formulate arithmetical laws such as $\begin{aligned} a + b = b + a \end{aligned}$ that apply to all real numbers in place of $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ .
::公式计算法,例如 a+b=b+a,适用于代替a和b的所有实际数字。

Refer to “unknown” numbers. For instance: find a number $\begin{aligned} x \end{aligned}$ such that $\begin{aligned} 3 x + 1 = 10 \end{aligned}$ .
::参考“ 未知” 数字。例如: 找到一个数字x, 即 3x+1=10 。

Write more compactly about functional relationships such as, “If you sell $\begin{aligned} x \end{aligned}$ tickets, then your profit will be $\begin{aligned} 3 x - 10 \end{aligned}$ dollars, or $\begin{aligned} f (x) = 3 x - 10 \end{aligned}$ , where $\begin{aligned} f \end{aligned}$ is the profit function , and $\begin{aligned} x \end{aligned}$ is the input (how many tickets you sell).

::更严格地写下功能关系,比如,“如果你卖了x门票,那么你的利润将是3x-10美元,或者f(x)=3x-10,其中f是利润的函数,x是输入(有多少张门票你卖了)。

Writing an Algebraic Equation
::撰写代数方程式

Write an algebraic equation for the perimeter and area of the rectangle below.
::为以下矩形的周界和区域写一个代数方程。

To find the perimeter, we add the lengths of all 4 sides. We can still do this even if we don’t know the side lengths in numbers, because we can use variables like $\begin{aligned} l \end{aligned}$ and $\begin{aligned} w \end{aligned}$ to represent the unknown length and width. If we start at the top left and work clockwise, and if we use the letter $\begin{aligned} P \end{aligned}$ to represent the perimeter, then we can say:
::为了找到周界,我们加上四边的长度。即使我们不知道数字的侧长, 我们仍然可以这样做, 因为我们可以使用I和W等变量来代表未知的长度和宽度。如果我们从左上方开始,按时针工作, 如果我们用字母P来代表周界, 那么我们可以说:

$\begin{aligned} P = l + w + l + w \end{aligned}$

::P=l+w+l+w

Since we are adding two $\begin{aligned} l^{'} s \end{aligned}$ and two $\begin{aligned} w^{'} s \end{aligned}$ , we can say that:
::由于我们增加了两个l__和两个w_,我们可以说:

$\begin{aligned} P = 2 \cdot l + 2 \cdot w \end{aligned}$

::P=2l+2w

It's customary in algebra to omit multiplication symbols whenever possible. For example, $\begin{aligned} 11 x \end{aligned}$ means the same thing as $\begin{aligned} 11 \cdot x \end{aligned}$ or $\begin{aligned} 11 \times x \end{aligned}$ . We can therefore also write:
::代数中的习惯是尽可能省略乘数符号。例如, 11x 的意思与 11x 或 11x 或 11x 相同。因此, 我们也可以写 :

$\begin{aligned} P = 2 l + 2 w \end{aligned}$

::P=2l+2w

Area is length multiplied by width. In algebraic terms we get:
::面积乘以宽度。以代数值计算,我们得到:

$\begin{aligned} A = l \times w \to A = l \cdot w \to A = l w \end{aligned}$

::A=lxwA=lwA=lwA=lw

Note: $\begin{aligned} 2 l + 2 w \end{aligned}$ by itself is an example of a variable expression , whereas $\begin{aligned} P = 2 l + 2 w \end{aligned}$ is an example of an equation . The difference between expressions and equations is the presence of an equals sign (=).
::注: 2l+2w 本身是变量表达式的一个示例,而 P=2l+2w 是方程式的一个示例。表达式和方程式之间的差异是等号(=)的存在。

In the above example, we found the simplest possible ways to express the perimeter and area of a rectangle when we don’t yet know what its length and width actually are. Now, when we encounter a rectangle whose dimensions we do know, we can simply substitute (or "plug in") those values in the above equations. In Algebra, you will encounter many expressions that can be evaluated by substituting values for the given variables.
::在上述例子中,当我们还不知道矩形的长度和宽度实际上是什么时,我们找到了表达矩形周界和面积的最简单方法。现在,当我们遇到一个我们确实知道其维度的矩形时,我们可以简单地替代(或“插入 ” ) 上述方程式中的值。在代数中,您会遇到许多可以通过替换给定变量的值来评估的表达式。

Writing Equations Involving Variables
::涉及变量的书写等量

1. Eric has some money in his savings account. How much more money does he need in order to buy a game that costs $98?
::1. Eric在储蓄账户里存了一些钱,他还需要多少钱才能买到价值98美元的游戏?

Let $\begin{aligned} M \end{aligned}$ represent the money that Eric still needs, and let $\begin{aligned} S \end{aligned}$ be the money that he has in his savings account. Then, by subtracting the money he already has from the total cost of the game, we can figure out how much money he still needs:
::让M代表Eric仍然需要的钱,让S代表他储蓄账户里的钱。然后,通过从游戏总成本中减去他已经拥有的钱,我们就能知道他还需要多少钱:

$\begin{aligned} M = 98 - S \end{aligned}$

::M=98-S

2. Write an equation for the sum of 3 times some number and 5.
::2. 写一个方程,总和为若干数字和5的3倍和5。

Let $\begin{aligned} S \end{aligned}$ be the total sum. Let $\begin{aligned} n \end{aligned}$ be " some number." Then " 3 times some number" is $\begin{aligned} 3 \cdot n \end{aligned}$ and then the sum of $\begin{aligned} 3 \cdot n \end{aligned}$ and 5 is:
::以总和计数。以“ 某个数字 ” 。然后, “ 3乘以3 ” 是 3 , 3 和 5 的总和是 :

$\begin{aligned} S = 3 n + 5 \end{aligned}$

::S=3n+5

Watch this video for help with the Examples above.
::观看此视频, 帮助了解上面的例子。

Example
::示例示例示例示例

Example 1
::例1

Alex has a certain amount of nickels and dimes in a jar. Write an algebraic equation for how much money she has, in terms of the number of nickels and dimes.
::亚历克斯在罐子里有一定数量的镍和硬币。写一个代数方程式, 说明她有多少钱, 就镍和硬币的数量而言。

Let $\begin{aligned} n \end{aligned}$ be the number of nickels, and $\begin{aligned} d \end{aligned}$ be the number of dimes, that Alex has in the jar. Since each nickel is worth $0.05, the amount of money she has in nickels will be $\begin{aligned} 0.05 times n, or 0.05 n . \end{aligned}$
::以亚历克斯在罐子里的硬币数量计, 以亚历克斯在罐子里的硬币数量计。因为每个硬币的价值是0.05美元, 她的硬币数量将是0.05乘以n, 或0.05n。

Since each dime is worth $0.10, the amount of money she has in dimes will be $\begin{aligned} 0.10 d . \end{aligned}$
::因为每分钱都值10美元她的零分钱是0.10美元

This means that the total amount of money, $\begin{aligned} M \end{aligned}$ , that Alex has is $\begin{aligned} M = 0.05 n + 0.10 d . \end{aligned}$
::这意味着亚历克斯的货币总额M是M=0.05n+0.10d。

$\begin{aligned} M = 0.05 n + 0.10 d \end{aligned}$

::M=0.05n+0.10d

Review
::回顾

For 1-4, write the following in a more condensed form by leaving out the multiplication symbol.
::1-4时,将乘数符号删除,以更压缩的形式写下以下文字。

$\begin{aligned} 2 \times y \end{aligned}$
::2×y

$\begin{aligned} 1.35 \cdot y \end{aligned}$
::1.35y

$\begin{aligned} 3 \frac{1}{4} \times m \end{aligned}$
::314×m

$\begin{aligned} \frac{1}{4} \cdot z \end{aligned}$
::14z

For 5-10, write an equation for the following situations.
::5 -10, 写一个方程式, 描述下列情况。

The amount of money Andrea has in a jar full of quarters and dimes.
::Andrea在一罐里装满了硬币和硬币的钱

The amount of money Michelle has in her coin purse if it only contains quarters, dimes and pennies.
::蜜雪儿的硬币袋里有多少钱如果里面只有零钱、零钱和零钱

The sum of 7 and 6 times some number.
::总和7和6乘以一些数字。

4 less than 20 times some number.
::4个小于20倍的某个数字。

The amount of money you will earn if you are paid $10.25 an hour and spend $4.00 round trip to get to and from work.
::如果你得到每小时10.25美元的工资,并花4.00美元往返工作,你将挣得的金额。

A father earns a $2000 dividend from an oil investment and distributes it equally amongst his children.
::父亲从石油投资中赚取2 000美元的红利,并在子女中平等分配。

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。

章节大纲

常规

变量变量的定义

Variables ::变量变量变量

Writing an Algebraic Equation ::撰写代数方程式

Writing Equations Involving Variables ::涉及变量的书写等量

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Variables
::变量变量变量

Writing an Algebraic Equation
::撰写代数方程式

Writing Equations Involving Variables
::涉及变量的书写等量

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)