Section: 包含一个或多个变量的表达式 | 1.5包含一个或多个变量的表达式

Section outline

Lisa's older sister Theresa has agreed to loan her some money so she can buy a new pair of jeans. Theresa is going to charge Lisa interest until Lisa pays her back. Theresa tells Lisa that she will calculate how much interest Lisa owes after $\begin{aligned} t \end{aligned}$ months according to the expression $\begin{aligned} \frac{1}{10} \cdot j \cdot t \end{aligned}$ where $\begin{aligned} j \end{aligned}$ is the cost of the jeans. Lisa wants to figure out how much interest she will owe Theresa if the jeans cost $60 and she doesn't pay Theresa back for 3 months.
::Lisa的姐姐Theresa同意借给她一些钱,以便她能够买到一条新牛仔裤。Theresa将收取Lisa的利息,直到Lisa还给她。Theresa告诉Lisa,她将计算Lisa在12个月之后的利息。根据1 10 j j t的表达方式,这牛仔裤的成本是j。Lisa想弄清楚,如果牛仔裤花费60美元,3个月不还Tresa,她会欠Tresa多少利息。

In this concept, you will learn how to evaluate expressions with more than one variable .
::在此概念中,您将学习如何用多个变量来评价表达式。

Evaluating Expressions with One or More Variables
::带有一个或多个变量的评价表达式

A variable is a symbol or letter (such as $\begin{aligned} x, m, R, y, P, \end{aligned}$ or $\begin{aligned} a \end{aligned}$ ) that is used to represent a quantity that might change in value. A variable expression is a mathematical phrase that contains numbers, operations , and variables.
::变量是一个符号或字母(例如 x、 m、 R、 y、 P 或 a),用来表示数值可能变化的数量。变量表达式是一个包含数字、操作和变量的数学短语。

Here are some examples of variable expressions:
::以下是一些变量表达式的例子:

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

$\begin{aligned} 10 r - x \end{aligned}$
::10 r- xx

$\begin{aligned} b^{3} + 2 \end{aligned}$
::b 3+2 b 3+2

$\begin{aligned} m x - 4 \end{aligned}$
::m x - 4 个

You can use a variable expression to describe a real world situation where one or more quantities has an unknown value or can change in value.
::您可以使用变量表达式来描述一个或一个以上数量值未知或值可能变化的真实世界状况。

To evaluate a variable expression means to find the value of the expression for given values of the variables. To evaluate, substitute the given values for the variables in the expression and simplify using the . To follow the order of operations, you always need to do any multiplication/ division first before any addition/ subtraction .
::要对变量表达式进行评价, 需要为变量的指定值查找表达式的值。要评估, 请将给定值替换为表达式中的变量, 并简化使用。要遵循操作的顺序, 您总是需要在增加/ 减法之前先做任何乘/ division 。

Here is an example.
::举一个例子。

Evaluate the expression $\begin{aligned} x y + x \end{aligned}$ if $\begin{aligned} x = 2 \end{aligned}$ and $\begin{aligned} y = 4 \end{aligned}$ .
::如果 x = 2, y = 4, 则评价表达式 x y + x。

First, substitute 2 in for the letter $\begin{aligned} x \end{aligned}$ and 4 in for the letter $\begin{aligned} y \end{aligned}$ in the expression.
::首先,将字母 x 中的 2 替换为字母 y 中的 4 替换为表达式中的 y 。

$\begin{aligned} (2) (4) + 2 \end{aligned}$

Notice that you can put " data-term="Parentheses" role="term" tabindex="0"> parentheses around the numbers to keep them separate and to indicate multiplication .
::请注意,您可以在数字周围加括号,以保持数字分开,并注明乘法。

Now, simplify the expression using the order of operations. You will need to multiply first and then add.
::现在, 使用操作顺序简化表达式。您需要先乘再添加。

$\begin{aligned} \begin{array}{rcl} (2) (4) + 2 & = & 8 + 2 \\ = & 10 \end{array} \end{aligned}$

The answer is 10.
::答案是10个

Here is another example that involves fractions.
::下面是另一个涉及分数的例子。

Evaluate the expression $\begin{aligned} m x + 3 m \end{aligned}$ for $\begin{aligned} x = \frac{2}{3} \end{aligned}$ and $\begin{aligned} m = 9 \end{aligned}$ .
::评价表达式m x + 3m x x = 2 3 和 m = 9。

First, substitute $\begin{aligned} \frac{2}{3} \end{aligned}$ in for the letter $\begin{aligned} x \end{aligned}$ and 9 in for the letter $\begin{aligned} m \end{aligned}$ in the expression.
::第一,以字母x取代字母2 3,以字母m代替字母x,以字母m代替字母x9。

$\begin{aligned} (9) (\frac{2}{3}) + 3 (9) \end{aligned}$

Now, simplify the expression using the order of operations. You will need to multiply each part of the expression first.
::现在, 使用操作顺序简化表达式。您需要先将表达式的每个部分乘以。

Remember, to multiply a whole number times a fraction , you can first turn the whole number into a fraction by writing it over 1. Then, multiply the numerators and multiply the denominators. Finally, simplify the fraction.
::记住,要将一个整数乘以一个分数,您可以首先将整个数乘以一个分数,然后通过写入 1 将整数转换成一个分数。然后,将数乘以数字,再乘以分数。最后,简化分数。

$\begin{aligned} \begin{array}{rcl} (9) (\frac{2}{3}) + 3 (9) & = & \frac{9}{1} \cdot \frac{2}{3} + 27 \\ = & \frac{18}{3} + 27 \\ = & 6 + 27 \end{array} \end{aligned}$

Next, continue to simplify the expression using the order of operations by adding the two terms .
::其次,继续使用操作顺序简化表达方式,增加这两个词。

$\begin{aligned} 6 + 27 = 33 \end{aligned}$

The answer is 33.
::答案是33岁

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Lisa and the new pair of jeans she wants to buy.
::早些时候,你得到一个问题关于丽莎和新牛仔裤她想买。

Her sister Theresa will loan her the money to buy them, but she will need to pay her sister back plus interest. Her sister will use the following expression to calculate how much interest Lisa owes her:
::她的妹妹Theresa会借钱给她买,但她需要偿还她妹妹的利息。她妹妹会用以下的表达方式计算Lisa欠她多少利息:

$\begin{aligned} \frac{1}{10} \cdot j \cdot t \end{aligned}$

In this expression, $\begin{aligned} j \end{aligned}$ represents the cost of the jeans and $\begin{aligned} t \end{aligned}$ represents the number of months before Lisa pays Theresa back.
::j表示牛仔裤的费用, t表示Lisa还给Theresa之前的月数。

Lisa wants to figure out how much interest she will owe Theresa if the jeans cost $60 and she pays Theresa back in 3 months.
::Lisa想弄清楚如果牛仔裤花了60美元她会欠Tesesa多少利息

First, Lisa should substitute 60 in for the letter $\begin{aligned} j \end{aligned}$ and 3 in for the letter $\begin{aligned} t \end{aligned}$ in the expression.
::首先,Lisa应以60取代j,3代替t。

$\begin{aligned} \frac{1}{10} \cdot 60 \cdot 3 \end{aligned}$

Now, simplify the expression using the order of operations. You will need to multiply $\begin{aligned} \frac{1}{10} \end{aligned}$ by 60 and then multiply that result by 3.
::现在,使用操作顺序简化表达式。您需要乘以 110 乘以 60, 然后乘以结果 3 。

Remember, to multiply a whole number and a fraction, you can first turn the whole number into a fraction by writing it over 1. Then, multiply the numerators and multiply the denominators. Finally, simplify the fraction.
::记住,要乘以整数和分数,您可以先将整数写成1,然后将数乘数乘数乘数乘数乘数。最后,简化分数。

$\begin{aligned} \begin{array}{rcl} \frac{1}{10} \cdot 60 \cdot 3 & = & \frac{1}{10} \cdot \frac{60}{1} \cdot 3 \\ = & \frac{60}{10} \cdot 3 \\ = & 6 \cdot 3 \\ = & 18 \end{array} \end{aligned}$

The answer is that Lisa will owe Theresa $18 in interest if she waits 3 months to pay Theresa back. This means in 3 months she will owe her sister a total of $\begin{aligned} 60 + 18 = $ 78 \end{aligned}$ for the jeans and the interest!
::答案是丽莎如果等了三个月才还回特蕾莎,就会欠特蕾莎18美元利息。这意味着三个月后,她将欠妹妹60+18=78美元的牛仔裤和利息!

Example 2
::例2

Evaluate the expression $\begin{aligned} x y + x y \end{aligned}$ for $\begin{aligned} x = \frac{1}{2} \end{aligned}$ and $\begin{aligned} y = \frac{2}{3} \end{aligned}$ .
::以 x = 1 2 和 y = 2 3 来评价表达式 x y + x y。

First, substitute $\begin{aligned} \frac{1}{2} \end{aligned}$ in for the letter $\begin{aligned} x \end{aligned}$ and $\begin{aligned} \frac{2}{3} \end{aligned}$ in for the letter $\begin{aligned} y \end{aligned}$ in the expression.
::首先,替换字母x的12和字母y的2 3。

$\begin{aligned} (\frac{1}{2}) (\frac{2}{3}) + (\frac{1}{2}) (\frac{2}{3}) \end{aligned}$

Now, simplify the expression using the order of operations. You will need to multiply each part of the expression first.
::现在, 使用操作顺序简化表达式。您需要先将表达式的每个部分乘以。

Remember, to multiply fractions you should multiply the numerators and multiply the denominators. Then, simplify the fraction.
::记住, 要乘以分数, 您应该乘以数解器, 乘以分母。然后简化分数。

$\begin{aligned} \begin{array}{rcl} (\frac{1}{2}) (\frac{2}{3}) + (\frac{1}{2}) (\frac{2}{3}) & = & \frac{2}{6} + \frac{2}{6} \\ = & \frac{1}{3} + \frac{1}{3} \end{array} \end{aligned}$

Next, continue to simplify the expression using the order of operations by adding the two terms.
::其次,继续使用操作顺序简化表达方式,增加这两个词。

Remember, to add fractions you need a common denominator. Here, you already have a common denominator of 3. So, add the numerators of your fractions and keep the common denominator as your denominator.
::记住, 要添加分数, 您需要一个共同分母。在这里, 您已经有一个 3 的共同分母。所以, 请添加您分数的数解器, 并保留共同分母作为您的分母。

$\begin{aligned} \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \end{aligned}$

The answer is $\begin{aligned} \frac{2}{3} \end{aligned}$ .
::答案是2 3。

Example 3
::例3

Evaluate the expression $\begin{aligned} a b + 7 \end{aligned}$ when $\begin{aligned} a \end{aligned}$ is 9 and $\begin{aligned} b \end{aligned}$ is 8.
::如果a为9,b为8,则评价a b + 7的表达式。

First, substitute 9 in for the letter $\begin{aligned} a \end{aligned}$ and 8 in for the letter $\begin{aligned} b \end{aligned}$ .
::第一,用字母A中的9取代字母b中的8代替字母b。

$\begin{aligned} (9) (8) + 7 \end{aligned}$

Now, simplify the expression using the order of operations. You will need to multiply first and then add.
::现在, 使用操作顺序简化表达式。您需要先乘再添加。

$\begin{aligned} \begin{array}{rcl} (9) (8) + 7 & = & 72 + 7 \\ = & 79 \end{array} \end{aligned}$

The answer is 79.

Example 4
::例4

Evaluate the expression $\begin{aligned} x y + z x \end{aligned}$ when $\begin{aligned} x \end{aligned}$ is 2, $\begin{aligned} y \end{aligned}$ is 5, and $\begin{aligned} z \end{aligned}$ is 7.
::以 x 2 y 5 和 z 7 来评价表达式 x y + z x 。

First, substitute 2 in for the letter $\begin{aligned} x \end{aligned}$ , 5 in for the letter $\begin{aligned} y \end{aligned}$ , and 7 in for the letter $\begin{aligned} z \end{aligned}$ .
::首先,替换字母x 2,字母y 5,字母z 7。

$\begin{aligned} (2) (5) + (7) (2) \end{aligned}$

Now, simplify the expression using the order of operations. You will need to multiply each part of the expression first and then add.
::现在, 使用操作顺序简化表达式。您需要先将表达式的每个部分乘以, 然后添加。

$\begin{aligned} \begin{array}{rcl} (2) (5) + (7) (2) & = & 10 + 14 \\ = & 24 \end{array} \end{aligned}$

The answer is 24.

Example 5
::例5

Evaluate the expression $\begin{aligned} x y + x \end{aligned}$ when $\begin{aligned} x \end{aligned}$ is $\begin{aligned} \frac{1}{4} \end{aligned}$ and $\begin{aligned} y \end{aligned}$ is 3.
::以 x 14 和 y 3 来评价表达式 x y + x。

First, substitute $\begin{aligned} \frac{1}{4} \end{aligned}$ in for the letter $\begin{aligned} x \end{aligned}$ and 3 in for the letter $\begin{aligned} y \end{aligned}$ in the expression.
::首先,替换字母x的14和字母y的3。

$\begin{aligned} (\frac{1}{4}) (3) + \frac{1}{4} \end{aligned}$

Now, simplify the expression using the order of operations. You will need to multiply the first part of the expression first.
::现在, 使用操作顺序简化表达式。您需要先乘以表达式的第一部分。

Remember, to multiply a whole number and a fraction, you can first turn the whole number into a fraction by writing it over 1. Then, multiply the numerators and multiply the denominators.
::记住,要乘以一个整数和一个分数,您可以先将整数写成1,然后将数乘数乘以数分数,然后将分数乘以数分数,然后将整数转换成一个分数。

$\begin{aligned} \begin{array}{rcl} (\frac{1}{4}) (3) + \frac{1}{4} & = & \frac{1}{4} \cdot \frac{3}{1} + \frac{1}{4} \\ = & \frac{3}{4} + \frac{1}{4} \end{array} \end{aligned}$

Next, continue to simplify the expression using the order of operations by adding the two terms.
::其次,继续使用操作顺序简化表达方式,增加这两个词。

Remember, to add fractions you need a common denominator. Here, you already have a common denominator of 4. So, add the numerators of your fractions and keep the common denominator as your denominator. Then, simplify.
::记住, 要添加分数, 您需要一个共同的分母。在这里, 您已经有一个 4 的共同分母。所以, 请添加您分数的分子, 并保留共同分母作为您的分母。然后, 简化。

$\begin{aligned} \begin{array}{rcl} \frac{3}{4} + \frac{1}{4} & = & \frac{4}{4} \\ = & 1 \end{array} \end{aligned}$

The answer is 1.
::答案是 1 。

Review
::回顾

Evaluate each expression if $\begin{aligned} x = 3 \end{aligned}$ and $\begin{aligned} y = 4 \end{aligned}$ .
::如果 x = 3 和 y = 4,则评价每个表达式。

$\begin{aligned} 3 y + 9 x \end{aligned}$
::3y + 9x 3y + 9x

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

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

Evaluate each expression if $\begin{aligned} x = 10 \end{aligned}$ and $\begin{aligned} y = 5 \end{aligned}$ .
::如果 x = 10 和 Y = 5,评价每个表达式。

$\begin{aligned} 5 x + x y \end{aligned}$
::5x+xy 5x+xy

$\begin{aligned} 3 x + 2 y \end{aligned}$
::3 x + 2 yy 3 x + 2 y

$\begin{aligned} 2 x + 3 y \end{aligned}$
::2 x + 3 yx 2 x + 3y

$\begin{aligned} 4 x + 3 y + x \end{aligned}$
::4 x + 3 y + x 4 x + 3 y + x

$\begin{aligned} 5 x + 3 y + 2 x \end{aligned}$
::5 x + 3 y + 2 x 5 x + 3 y + 2 x

Evaluate each expression if $\begin{aligned} y = 2 \end{aligned}$ and $\begin{aligned} z = 4 \end{aligned}$ .
::如果y=2和z=4,评价每个表达式。

$\begin{aligned} 2 y + 3 y - 2 z \end{aligned}$
::2 y + 3 y y - 2 z

$\begin{aligned} 5 y + 3 y - 3 z \end{aligned}$
::5 y + 3 y y - 3 z 5 y + 3 y y - 3 z

$\begin{aligned} 2 y + 5 y - 2 z \end{aligned}$
::2y + 5y + 5y - 2 z

$\begin{aligned} 4 y + 3 y - 2 z \end{aligned}$
::4y + 3y + 3y - 2 z

Evaluate $\begin{aligned} 6 z - 2 (z + x) \end{aligned}$ if $\begin{aligned} x \end{aligned}$ is 3 and $\begin{aligned} z \end{aligned}$ is 4.
::如果x为3,则评价6z-2(z+x),如果z为4,则评价6z-2(z+x)。

Evaluate $\begin{aligned} 8 a + 3 b - 2 c \end{aligned}$ if $\begin{aligned} a \end{aligned}$ is 5, $\begin{aligned} b \end{aligned}$ is 4 and $\begin{aligned} c \end{aligned}$ is 3.
::评价8 a+3 b-2 c,如果a为5 b为4,c为3。

Section outline

Evaluating Expressions with One or More Variables ::带有一个或多个变量的评价表达式

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

Evaluating Expressions with One or More Variables
::带有一个或多个变量的评价表达式

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