Course: 1.4 评价代数表达式和等同

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评价代数表达式和等同
Your summer landscaping job pays a fixed rate of $20 per job plus $4 an hour. How much total money would you would make if it takes you 3 hours to complete a single job?
::你暑期的园林美化工作每份工作支付20美元固定费率,每份工作加每小时4美元。如果完成一份单一工作需要3小时,你会赚多少钱?

Evaluating Algebraic Expressions and Equations
::评价代数表达式和等同

You have probably seen letters in a mathematical expression , such as $\begin{align*}3x-8\end{align*}$ . These letters, also called variables, represent an unknown number. One of the goals of algebra is to solve various equations for a variable . Typically, $\begin{align*}x\end{align*}$ is used to represent the unknown number, but any letter can be used.
::您可能已经看到数学表达式中的字母, 如 3x-8 。这些字母, 也称为变量, 代表未知的数字。代数的目标之一是解析变量的不同方程式。通常, x 用来代表未知的数字, 但任何字母都可以使用。

To evaluate an expression or equation , we would need to substitute in a given value for the variable and test it. In order for the given value to be true for an equation, the two sides of the equation must simplify to the same number.
::为了评价表达式或方程,我们需要用给定值来替代变量并测试变量。要使给定值对方程式真实,方程式的两面必须简化为相同数字。

Let's evaluate the following expressions and equations.
::让我们来评估以下表达式和方程式。

Evaluate $\begin{align*}2x^2-9\end{align*}$ for $\begin{align*}x=-3\end{align*}$ .
::x% 3 的 2x2- 9 评估。

We know that $\begin{align*}2x^2-9\end{align*}$ is an expression because it does not have an equals sign. Therefore , to evaluate this expression, plug in -3 for $\begin{align*}x\end{align*}$ and simplify using the .
::我们知道 2x2- 9 是一个表达式, 因为它没有等号。因此, 要评价此表达式, 请插入 - 3 , x 并简化使用。

$\begin{aligned} 2 (- 3)^{2} - 9 \to (- 3)^{2} = - 3 \cdot - 3 = 9 \\ 2 (9) - 9 \\ 18 - 9 \\ 9 \end{aligned}$
$\begin{align*}&2(-3)^2-9 \rightarrow (-3)^2=-3 \cdot -3=9\\ &2(9)-9\\ &18-9\\ &9\end{align*}$

You will need to remember that when squaring a negative number, the answer will always be positive. There are three different ways to write multiplication : $\begin{align*}2 \times 9, 2 \cdot 9\end{align*}$ , and 2(9).
::您需要记住, 当计算负数时, 答案总是正数。写乘法有三种不同的方式 : 2x9, 2x9, 2x9, 和 2( 9) 。

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

Even though the directions are different, this problem is almost identical to #1 above. However, this is an equation because of the equals sign. Both sides of an equation must be equal to each other in order for it to be true. Plug in 5 everywhere there is an $\begin{align*}x\end{align*}$ . Then, determine if both sides are the same.
::即使方向不同, 这个问题与上面的 # 1 几乎完全相同。但是, 这是因等号而异的方程。等式的两面必须等对齐, 才能让其真实性成为事实。任何地方的 5 点中都有一个 x。然后, 确定双方是否相同。

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

Because $\begin{align*}4 \ne 14\end{align*}$ , 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。

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

Here, $\begin{align*}t\end{align*}$ is the variable and it is listed twice in this equation. Plug in -2 everywhere there is a $\begin{align*}t\end{align*}$ and simplify.
::这里, t 是变量, 它在此方程中列出两次。在 - 2 中插入 t 和简化。

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

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

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about your summer landscaping job.
::早些时候,你得到一个问题关于你的暑期园林美术工作。

Rewrite the sentence as an algebraic expression . $20 plus $4 an hour, would be $\begin{align*}20+4h\end{align*}$ , where h equals the number of hours you work. Then, evaluate the expression for $\begin{align*}h = 3\end{align*}$ .
::将句子重写为代数表达式。 20美元加每小时4美元, 将是 20+4h, 其中 h 等于您工作的小时数。然后, 评估 h=3 的表达式。

$\begin{align*}20+4(3)=20+12=32\end{align*}$

You will make a total of $32 for this particular job.
::这份工作总共需要32美元

Example 2
::例2

Evaluate $\begin{align*}s^3-5s+6\end{align*}$ for $\begin{align*}s=4\end{align*}$ .
::评价s=4的S3-5s+6。

Plug in 4 everywhere there is an $\begin{align*}s\end{align*}$ .
::插在4, 任何地方有一个s。

$\begin{aligned} 4^{3} - 5 (4) + 6 \\ 64 - 20 + 6 \\ 50 \end{aligned}$
$\begin{align*}&4^3-5(4)+6\\ &64-20+6\\ &50\end{align*}$

Example 3
::例3

Determine if $\begin{align*}a=-1\end{align*}$ is a solution to $\begin{align*}4a-a^2+11=-2-2a\end{align*}$ .
::确定 a% 1 是 4a - a2+11+% 2-2a 的解决方案。

Plug in -1 for $\begin{align*}a\end{align*}$ and see if both sides of the equation are the same.
::插在 - 1 中, 检查方程式的两侧是否相同。

$\begin{aligned} ? \\ 4 (- 1) - (- 1)^{2} + 11 & = - 2 - 2 (- 1) \\ ? \\ - 4 - 1 + 11 & = - 2 + 2 \\ 6 & \neq 0 \end{aligned}$
$\begin{align*}& \ ? \\ 4(-1)-(-1)^2+11 &= -2-2(-1)\\ & \ ? \\ -4-1+11 &=-2+2\\ 6 & \neq 0\end{align*}$

Because the two sides are not equal, -1 is not a solution to this equation.
::由于双方不平等, -1不是解决这一方程式的办法。

Review
::回顾

Evaluate the following expressions for $\begin{align*}x = 5\end{align*}$ .
::x=5 的表达式评价如下:

$\begin{align*}4x-11\end{align*}$
::4-11 4x-11

$\begin{align*}x^2+8\end{align*}$
::x2+8x2+8

$\begin{align*}\frac{1}{2}x+1\end{align*}$
::12x+1 12x+1

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

$\begin{align*}-2a+7; a=-1\end{align*}$
::- 2a+7;a%1

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

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

$\begin{align*}x^2-5x+6; x=3\end{align*}$
::x2 - 5x+6;x=3

$\begin{align*}8p^2-3p-15; p=-2\end{align*}$
::8p2-3p- 15;p%2

$\begin{align*}m^3-1; m=1\end{align*}$
::m3- 1; m= 1

Determine if the given values are solutions to the equations below.
::确定给定值是否为以下方程式的解决方案。

$\begin{align*}x^2-5x+4=0; x=4\end{align*}$
::x2 - 5x+4=0;x=4

$\begin{align*}y^3-7=y+3; x=2\end{align*}$
::y3+3;x=2 y3 - 7=y+3;x=2

$\begin{align*}7x-3=4; x=1\end{align*}$
::7x-3=4;x=1

$\begin{align*}6z+z-5=2z+12;z=-3\end{align*}$
::6z+z-5=2z+12;z3

$\begin{align*}2b-5b^2+1=b^2;b=6\end{align*}$
::2-5b2+1=b2;b=6

$\begin{align*}-\frac{1}{4}g+9=g+15;g=-8\end{align*}$
::- 14g+9=g+15;g8

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

$\begin{align*}ab-c\end{align*}$
::ab-c

$\begin{align*}b^2+2d\end{align*}$
::b2+2d (b2+2d)

$\begin{align*}c+\frac{1}{2}b-a\end{align*}$
::c+12b-a

$\begin{align*}b(a+c)-d^2\end{align*}$
::b (a+c)-d2

For problems 20-25, use the equation $\begin{align*}y^2+y-12=0\end{align*}$ .
::对于 20- 25 的问题, 请使用 y2+y- 12=0 方程式。

Is $\begin{align*}y = 4\end{align*}$ a solution to this equation?
::y=4是这个方程式的解决方案吗?

Is $\begin{align*}y = -4\end{align*}$ a solution to this equation?
::你们4是这个方程式的解决方案吗?

Is $\begin{align*}y = 3\end{align*}$ a solution to this equation?
::y=3是这个方程式的解决方案吗?

Is $\begin{align*}y = -3\end{align*}$ a solution to this equation?
::你们是这个方程式的解决方案吗?

Do you think there are any other solutions to this equation, other than the ones found above?
::你认为除了上面找到的方程式之外还有别的解决方案吗?

Challenge Using the solutions you found from problems 20-23, find the sum of these solutions and their product. What do you notice?
::使用你从20-23问题中找到的解决方案, 找到这些解决方案及其产品的总和。您注意到什么了 ?

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

General

评价代数表达式和等同

Evaluating Algebraic Expressions and Equations ::评价代数表达式和等同

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Evaluating Algebraic Expressions and Equations
::评价代数表达式和等同

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)