课程： 1.13 评价代数表达式

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评价代数表达式
Osmotic pressure is a measure of the tendency of a solution to take in water by a process called osmosis. Osmosis is a process where water passes though the boundary between the two liquids. The formula for calculating the osmotic pressure is $\begin{aligned} π = M R T \end{aligned}$ , where $\begin{aligned} π \end{aligned}$ is the pressure between the water and the solution, $\begin{aligned} M \end{aligned}$ is the molar concentration (essentially the concentration of the solution), $\begin{aligned} R \end{aligned}$ is the gas constant , and $\begin{aligned} T \end{aligned}$ is the temperature in Kelvin. ¹
::骨质压力是测量溶液通过一个称为渗透过程进入水中的趋势的尺度。骨质疾病是一个过程, 水经过两个液体之间的边界。计算水压的公式是 MRT, 是水与溶液之间的压力, M是摩尔浓度( 基本上是溶液的浓度), R是气体常数, T是克尔文的温度。 1

If we know the values of M , R , and T, we can find the osmotic pressure for a given solution. In this section, we will cover evaluating algebraic expressions like the expression in this example.
::如果我们知道 M、 R 和 T 的值, 我们就能找到给定解决方案的表面压力。在本节中, 我们将覆盖对像此示例中的代数表达式这样的代数表达式的评价。

Evaluating Algebraic Expressions
::评价代数表达式

A mathematical expression is a combination of numbers, letters, and/or operation symbols. For example, $\begin{aligned} 3 x - 8 \end{aligned}$ . The letters are called variables and they represent unknown numbers. The parts of the expression that are separated by addition and subtraction symbols are referred to as terms ; here, $\begin{aligned} 3 x \end{aligned}$ and 8 are terms. The numbers we multiply the variables by are called the coefficients . In this case, 3 is the coefficient of x .
::数学表达式是数字、字母和/或操作符号的组合。例如, 3x- 8 。字母被称为变量, 它们代表未知的数字。添加符号和减号分隔的表达式部分被称为术语; 这里, 3x 和 8 是术语。我们乘以变量的数值被称为系数。在这种情况下, 3 是 x 的系数。

An expression that includes variables is referred to as an algebraic expression or a variable expression . Although $\begin{aligned} x \end{aligned}$ is commonly used to represent an unknown number in algebraic expressions themselves or algebraic expressions that appear in equations, any letter can be used.
::包含变量的表达式被称为代数表达式或变量表达式。虽然在代数表达式本身或方程式中出现的代数表达式中, x 通常用于表示未知的数字,但任何字母都可以使用。

To evaluate an expression, we substitute or replace the variable with a given value for the variable. To determine if a given value satisfies an equation , we evaluate the expressions on each side of the equal sign. W e substitute for the variable with the given value and see if the two sides of the equation equal the same number. If this is true, the value is considered a solution to the equation.
::要评价表达式, 我们用给定值替换或替换变量。要确定给定值是否满足方程式, 我们用给定值来评估相等符号的每个侧面的表达式。我们用给定值来替换变量, 并查看方程式的两边是否相等。如果这是事实, 则该值被视为公式的解决方案。

Example 1
::例1

Evaluate $\begin{aligned} 2 x^{2} - 9 \end{aligned}$ when $\begin{aligned} x = - 3 \end{aligned}$ .
::x% 3 时评价 2x2- 9 。

Solution: We know that $\begin{aligned} 2 x^{2} - 9 \end{aligned}$ is an expression because it does not have an equal sign. Therefore , to evaluate this expression, substitute -3 for $\begin{aligned} x \end{aligned}$ and simplify using the .
::解答: 我们知道 2x2- 9 是一个表达式, 因为它没有相同的符号。因此, 要评价这个表达式, 请将 - 3 替换为 x, 并简化使用。

$\begin{aligned} 2 x^{2} - 9 \\ 2 (- 3)^{2} - 9 & Recall: (- 3)^{2} = - 3 \cdot - 3 = 9 \\ = 2 (9) - 9 \\ = 18 - 9 \\ = 9 \end{aligned}$

::2x2-92(-3-3)2-9 回顾-3)233=9=2(9)-9=18-9=9

R emember that when squaring a negative number, the result will be positive. Also, note there are three different ways to write multiplication . In the third line above, we could have written $\begin{aligned} 2 \times 9, 2 \cdot 9 \end{aligned}$ , or 2(9).
::记住当计算负数时, 结果将是正数。另外, 注意有三种不同的写乘法。在以上第三行, 我们本可以写入 2x9, 2x9, 2x9, 或 2 2( 9) 。

by Mathispower4u demonstrates how to evaluate algebraic expressions.
::由 Mathispower4u 演示如何评价代数表达式。

Example 2
::例2

Determine if $\begin{aligned} x = 5 \end{aligned}$ is a solution to $\begin{aligned} 3 x - 11 = 14 \end{aligned}$ .
::确定 x=5 是 3x- 11=14 的解决方案。

Solution: This is an equation because of the equal sign. Both sides of an equation must be equal to each other in order for it to be true. We can substitute 5 in the expression on the left, and then determine if both sides are the same.
::解决方案 : 这是一个等式, 因为它是平等的标志。等式的两面必须等同, 才能成为真实。我们可以用左侧的表达方式替换 5 , 然后确定双方是否相同。

$\begin{aligned} ? \\ 3 (5) - 11 & = 14 \\ 15 - 11 & \neq 14 \\ 4 & \neq 14 \end{aligned}$

Because $\begin{aligned} 4 \neq 14 \end{aligned}$ , this is not a true equation. Therefore, 5 is not a solution.
::因为4+14,这不是一个真正的方程式。因此,5不是解决之道。

Example 3
::例3

Determine if $\begin{aligned} t = - 2 \end{aligned}$ is a solution to $\begin{aligned} 7 t^{2} - 9 t - 10 = 36 \end{aligned}$ .
::确定 t2 是 7t2- 9t- 10=36 的解决方案。

Solution: Here, $\begin{aligned} t \end{aligned}$ is the variable and it is listed twice in this equation. Substitute -2 everywhere there is a $\begin{aligned} t \end{aligned}$ and simplify.
::解答:在这里, t 是变量, 它在这个方程式中列出两次。替代 - 2 处有 t 和简化。

$\begin{aligned} ? \\ 7 (- 2)^{2} - 9 (- 2) - 10 & = 36 \\ ? \\ 7 (4) + 18 - 10 & = 36 \\ ? \\ 28 + 18 - 10 & = 36 \\ 36 & = 36 \end{aligned}$

-2 is a solution to this equation.
::-2是这个方程式的解决方案

by MathEase demonstrates how to determine if a given value is a solution for a n equation.
::通过 MathEase 演示如何确定给定值是否是方程式的解决方案。

Example 4
::例4

What is the osmotic pressure of a given solution whose concentration ( M ) is 1.92 mol/L and whose temperature ( T) is 273.16 K? The gas constant ( R) is 8.31 J/Kmol. ¹
::某一溶液的浓度(M)为1.92毫升/升,温度(T)为273.16KK?气体常数(R)为8.31J/Kmol, 其表面压力是什么?

Solution: In the introduction, we were provided with the following formula for calculating the osmotic pressure: $\begin{aligned} π = M R T \end{aligned}$ . To find the osmotic pressure, we need to substitute the given values in for M , R , and T .
::解答:在导言中,我们得到了以下计算表面压力的公式: MRT。要找到表面压力,我们需要用给定值代替M、R和T。

$\begin{aligned} π & = M R T \\ = 1.92 \cdot 8.31 \cdot 273.16 \\ = 4, 358.32 \frac{J}{L} \end{aligned}$

::MRT=1.9283173.16=4 358.32 JL

Summary
::摘要

To evaluate an expression, substitute the given value for the variable and perform the indicated operations.
::要评价表达式,用给定值替换变量,并进行指定的操作。

To determine if a value is a solution to an equation, evaluate the expressions on each side of the equal sign and determine if the left side of the equation equals the right side of the equation .
::要确定一个值是否是方程的解决方案, 评估相等符号两侧的表达式, 并确定方程的左侧是否等于方程的右侧。

Review
::回顾

Evaluate the following expressions for the given value.
::评估给定值的下列表达式。

1. $\begin{aligned} - 2 a + 7; a = - 1 \end{aligned}$
::1.-2a+7;a1

2. $\begin{aligned} 3 t^{2} - 4 t + 5; t = 4 \end{aligned}$
::2. 3t2-4t+5;t=4

3. $\begin{aligned} \frac{2}{3} c - 7; c = - 9 \end{aligned}$
::3. 23c-7;c9

4. $\begin{aligned} x^{2} - 5 x + 6; x = 3 \end{aligned}$
::4. x2 - 5x+6;x=3

5. $\begin{aligned} 8 p^{2} - 3 p - 15; p = - 2 \end{aligned}$
::5. 8p2-3p-15;p2

6. $\begin{aligned} s^{3} - 5 s + 6; s = 4 \end{aligned}$
::6. s3-5s+6;s=4

Find the value of each expression, given that $\begin{aligned} a = - 1, b = 2, c = - 4 \end{aligned}$ , and $\begin{aligned} d = 0 \end{aligned}$ .
::查找每个表达式的值, 如果 a+1, b=2, c=4, 和 d=0 。

7. $\begin{aligned} a b - c \end{aligned}$
::7. ab-c

8. $\begin{aligned} b^{2} + 2 d \end{aligned}$
::8.b2+2d

9. $\begin{aligned} c + \frac{1}{2} b - a \end{aligned}$
::9. c+12b-a

10. $\begin{aligned} b (a + c) - d^{2} \end{aligned}$
::10. (a+c) - d2

Explore More
::探索更多

1. Determine if the given values are solutions to the equations below.
::1. 确定给定值是否是以下方程的解决方案。

a. $\begin{aligned} x^{2} - 5 x + 4 = 0; x = 4 \end{aligned}$
::a. x2 - 5x+4=0;x=4

b. $\begin{aligned} 6 z + z - 5 = 2 z + 12; z = - 3 \end{aligned}$
::b. 6z+z-5=2z+12;z3

c. $\begin{aligned} 2 b - 5 b^{2} + 1 = b^{2}; b = 6 \end{aligned}$
::c. 2b-5b2+1=b2;b=6

2. Your summer landscaping job pays a fixed rate of $20 per job plus $4 an hour. The expression is $\begin{aligned} 20 + 4 h \end{aligned}$ . How much total money would you would make if it takes you 3 hours to complete a single job?
::2. 暑期园林美化工作每份工作支付固定费率20美元,加每小时4美元,表达方式是20+4小时。

3. In physics, an equation for the velocity of an object is $\begin{aligned} v = \sqrt{2 g h} \end{aligned}$ . Find the velocity if the acceleration $\begin{aligned} g = 9.8 \frac{m}{s^{2}} \end{aligned}$ and height h = 10 m . ³
::3. 在物理学中,物体速度的方程为 v=2gh。如果加速度g=9.8ms2和高度h=10m.3,则查找速度。

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try these interactives that reinforce the concepts explored in this section:
::尝试这些强化本节所探讨概念的交互作用 :

Re ferences
::参考参考资料

1. “Osmotic Pressure,” last edited June 20, 2017, .
::1. 2017年6月20日经编辑的 " 地面压力 " 。

2. “Energy of Falling Object,” .
::2. " 坠落物体的能源, " 。

章节大纲

常规

评价代数表达式

Evaluating Algebraic Expressions ::评价代数表达式

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

Re ferences ::参考参考资料

Evaluating Algebraic Expressions
::评价代数表达式

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX

Re ferences
::参考参考资料