Section: 变量(标准等量)溶解公式 | 2.5 变量的溶解等量(标准等量)

Section outline

Your friend from Spain is planning a trip to the United States in the spring. In the United States, we measure temperature in degrees Fahrenheit, but in Europe, they measure temperature in degrees Celsius. The equation that relates Fahrenheit and Celsius is $\begin{aligned} F = \frac{9}{5} C + 32 \end{aligned}$ . You want to convey to your friend that it will be about $\begin{aligned} 60^{\circ} F \end{aligned}$ , but they only understand Celsius units. What do you say?
::你的朋友来自西班牙,正计划在春天前往美国。在美国,我们测量温度以华氏度计,但在欧洲,温度以摄氏度计。法氏度和摄氏度的方程式是F=95C+32。您想告诉您的朋友,这将是大约60美分,但他们只懂摄氏单位。您认为呢?

In the previous sections, we discovered how to solve an equation for a variable and get a numeric result. In this section, we will use the same techniques to solve for a variable in terms of other variables.
::在前几节中, 我们发现如何解析变量的方程式, 并获得数字结果。在本节中, 我们将会使用相同的方法来解答其它变量中的变量。

lesson content

Solving Literal Equations
::解决文学等量

In this section, we discuss literal equations , equations that contains letters or literals, what we call variables. To solve a literal equation for a specific variable, you are going to use the same techniques you used in the previous three sections. Remember, variables represent numbers, so you can manipulate variables just as you would numbers.
::在此节中, 我们讨论字面方程, 包含字母或字面方程的方程, 我们称之为变量。要解决特定变量的字面方程, 您将会使用前三节中使用的相同技术。记住, 变量代表数字, 这样您就可以像您所统计的那样操作变量。

T his video by CK-12 demonstrates how to solve literal equations.
::CK-12的这段影片展示了如何解析字面方程式。

Example 1
::例1

You want to convey to your friend from Spain that the temperature will be about 60 degrees Fahrenheit during their trip to the United States. They only understand temperatures in Celsius units. How do you convey what the temperature will be?
::您想向您来自西班牙的朋友转达一下, 在美国旅行期间, 华氏温度大约为60度。他们只理解摄氏单位的温度。您如何表达温度 ?

Solution: If we solve for $\begin{aligned} C \end{aligned}$ in the equation $\begin{aligned} F = \frac{9}{5} C + 32 \end{aligned}$ , we will be able to find a temperature in degrees Celsius that your friend will understand.
::解答:如果我们在方程式F=95C+32中解答C, 我们就能找到一个温度, 以你朋友能理解的摄氏度。

\begin{aligned} F = \frac{9}{5} C + 32 \\ \underline{- 32 - 32} \\ F - 32 = \frac{9}{5} C \\ \frac{5}{9} (F - 32) = \frac{5}{9} \times \frac{9}{5} C \\ \frac{5}{9} (F - 32) = C \end{aligned}

::F=95C+32-32-32_F-32=95C59(F-32=59x95C59(F-32)=C

Now, that we have an equation with Celsius isolated, we can determine what you should tell your friend.
::现在我们用摄氏隔离方程式来决定你该告诉你朋友什么

\begin{aligned} C & = \frac{5}{9} (F - 32) \\ = \frac{5}{9} (60 - 32) \\ = \frac{5}{9} (28) \\ = \frac{140}{9} = 15.56 \end{aligned}

::C=59(F-32)=59(60-32)=59(28)=1409=15.56

You should tell your friend it will be about $\begin{aligned} {15.56}^{\circ} C \end{aligned}$ or $\begin{aligned} 16^{\circ} C \end{aligned}$ .
::你应该告诉你朋友大约15.56°C或16°C

by CK-12 presents a similar problem.
::CK-12 提出了类似的问题。

If you would like to learn more about converting between Celsius and Fahrenheit, you can try the PLIX at the end of this section .
::如果您想了解更多关于摄氏和华氏之间的转换,请在本节结尾处使用PLIX。

Example 2
::例2

You open a savings account that earns at an annual rate of 3%. You plan to spend the interest earned on a new blender. If the new blender costs $59.99, how much money needs to be in the account if you want to buy the blender with the interest you have earned in one year?
::您打开了一个年利率为3%的储蓄账户。您计划将所得利息花在一个新的搅拌机上。如果新的搅拌机花费59. 99美元, 如果您想用一年来赚取的利息购买搅拌机, 需要多少钱?

Solution: We will use the simple interest formula , $\begin{aligned} I = P r t \end{aligned}$ , where $\begin{aligned} I \end{aligned}$ is the interest earned, $\begin{aligned} P \end{aligned}$ is the principal or the amount invested, $\begin{aligned} r \end{aligned}$ is the annual interest rate, and $\begin{aligned} t \end{aligned}$ is time in years. If we solve for $\begin{aligned} P \end{aligned}$ in the formula, we have
::解决方案:我们将使用简单的利息公式I=Prt,其中我为所得利息,P是本金或投资金额,r是年利率, t是年时间。如果我们在公式中解决P,我们就有了

\begin{aligned} \frac{P r t}{r t} & = \frac{I}{r t} \\ P & = \frac{I}{r t} \end{aligned}

::Prtrt = IrtP = Irt

Now, we can substitute the values we know to find the amount of money you need to save
::现在,我们可以取代我们所知道的价值来找到你需要储蓄的钱

\begin{aligned} P & = \frac{59.99}{.03 \times 1} \\ = 1999.67 \end{aligned}

::P=59.99.03x1=1999.67

You need to have $1,999.67 in the savings account to buy the blender with the interest earned.
::你需要在储蓄账户里有1 999.67美元才能用赚来的利息买下搅拌机

Example 3
::例3

The area of a triangle is $\begin{aligned} A = \frac{1}{2} b h \end{aligned}$ , where $\begin{aligned} b \end{aligned}$ is the base of the triangle and $\begin{aligned} h \end{aligned}$ is the height. You know that the area of a triangle is $\begin{aligned} 60 {in}^{2} \end{aligned}$ and the base is 12 in. Find the height.
::三角形的区域是 A=12bh, b 是三角形的底部, h 是高度。您知道三角形的区域是 60 英寸2, 基数是 12 英寸。查找高度。

Solution: First, solve the equation for $\begin{aligned} h \end{aligned}$ .
::解答:第一,解答h的方程。

\begin{aligned} A & = \frac{1}{2} b h \\ 2 \times A & = 2 \times \frac{1}{2} b h \\ \frac{2 A}{b} & = \frac{b h}{b} \\ \frac{2 A}{b} & = h \end{aligned}

::A=12bh2xA=2x12bh2Ab=bb2Ab=h

Now, plug in what we know to find $\begin{aligned} h \end{aligned}$ . $\begin{aligned} h = \frac{2 (60)}{12} = \frac{120}{12} = 10 \end{aligned}$ . The height is 10 inches.
::现在,插入我们所知道的找到 h. h=2( 6012=12012=10)。高度为 10 英寸。

Example 4
::例4

Given the equation $\begin{aligned} \frac{2 b}{3} + 6 a - 4 = 8 \end{aligned}$ , find $\begin{aligned} b \end{aligned}$ when $\begin{aligned} a = 1 \end{aligned}$ and $\begin{aligned} a = - 2 \end{aligned}$ .
::根据 2b3+6a-4=8的方程,在a=1和a=2时找到b。

Solution: First, isolate $\begin{aligned} b \end{aligned}$ . Move both the $\begin{aligned} 6 a \end{aligned}$ and the 4 over to the other side by doing the inverse operations .
::解决办法:首先,隔离 b. 进行反向操作,将6a 和 4 移动到另一侧。

\begin{aligned} \frac{2 b}{3} + 6 a - 4 = 8 \\ \underline{- 6 a + 4 = + 4 - 6 a} \\ \frac{2 b}{3} = 12 - 6 a \end{aligned}

::2b3+6a-4=8-6a+44-6a_2b3=12-6a

Now, we have a fraction multiplied by $\begin{aligned} b \end{aligned}$ . To undo this, we must multiply by the reciprocal of $\begin{aligned} \frac{2}{3} \end{aligned}$ , which is $\begin{aligned} \frac{3}{2} \end{aligned}$ . Be careful here. We must multiply everything in the equation by the reciprocal.
::现在,我们有一个分数乘以 b。要解决这个问题,我们必须乘以23的对等乘以23, 也就是32。小心这里。我们必须把方程中的一切乘以对等。

\begin{aligned} \frac{2 b}{3} = 12 - 6 a \\ \frac{3}{2} (\frac{2 b}{3} = 12 - 6 a) \\ \frac{3}{2} \cdot \frac{2}{3} b = \frac{3}{2} \cdot 12 - \frac{3}{2} \cdot 6 a \\ b = 18 - 9 a \end{aligned}

::2b3=12-6a32(2b3=12-6a) 32_23b=3212-32-6ab=18-9a

Even though 9 can be factored out in the above equation, it is not a necessary step in solving for $\begin{aligned} b \end{aligned}$ . Now, we can do what the question asks, find $\begin{aligned} b \end{aligned}$ when $\begin{aligned} a = 1 \end{aligned}$ and -2. Plug in these values for $\begin{aligned} a \end{aligned}$ .
::尽管可以在上述方程中考虑到9,但这不是解决 b 的一个必要步骤。现在,我们可以按问题的要求做,当a=1和-2时找到b。

\begin{aligned} a & = 1 : b = 18 - 9 (1) = 18 - 9 = 9 \\ a & = - 2 : b = 18 - 9 (- 2) = 18 + 18 = 36 \end{aligned}

::a=1:b=18-9(1)=18-9=9a=2:b=18-9(-2)=18+18=36

by CK-12 presents a similar example.
::CK-12 提供了类似的例子。

Summary
::摘要

Treat variables in a literal equation like numbers in a linear equation.
::在字面方程式中处理变量,就像在线性方程式中处理数字一样。

Use the same techniques you have used for solving linear equations.
::使用你用来解析线性方程式的相同技术。

Review
::回顾

These formulas will be used for the questions below.
::这些公式将用于下列问题。

Area of a Triangle	$\begin{aligned} A = \frac{1}{2} b h \end{aligned}$	$\begin{aligned} A = area, b = base, h = height \end{aligned}$
Area of a Parallelogram	$\begin{aligned} A = b h \end{aligned}$	$\begin{aligned} A = area, b = base, h = height \end{aligned}$
Area of a Trapezoid	$\begin{aligned} A = \frac{1}{2} h (b_{1} + b_{2}) \end{aligned}$	$\begin{aligned} A = area, h = height, b_{1} = one base, b_{2} = other base \end{aligned}$
Perimeter of a Rectangle	$\begin{aligned} P = 2 l + 2 w \end{aligned}$	$\begin{aligned} P = perimeter, l = length, w = width \end{aligned}$
Circumference of a Circle	$\begin{aligned} C = 2 π r \end{aligned}$	$\begin{aligned} C = circumference, r = radius \end{aligned}$

Use the formulas above to solve for the indicated variable.
::使用上面的公式解析指定的变量。

Radius in the circumference of a circle
::圆圆环圈中的半径

Height in the area of a parallelogram
::平行图层的高度

Length in the perimeter of a rectangle
::长长在矩形圆形周围

Base in the area of a triangle
::三角形区域的基数

One base in the area of a trapezoid
::1个基地,位于一个捕鼠类区域

Solve for the given variable in the following equations.
::在以下方程式中为给定变量解决。

6. $\begin{aligned} \frac{4}{5} m + \frac{2}{3} n = 24 \end{aligned}$ ; solve for $\begin{aligned} m \end{aligned}$ .
::6. 45m+23n=24;M.

7. $\begin{aligned} \frac{4}{5} m + \frac{2}{3} n = 24 \end{aligned}$ ; solve for $\begin{aligned} n \end{aligned}$ .
::7. 45m+23n=24;n.

For questions 8-9, use the formulas in this section to answer the following questions.
::对于问题8-9,使用本节中的公式回答下列问题。

8. If the temperature is $\begin{aligned} 86^{\circ} F \end{aligned}$ , what is it in Celsius?
::8. 如果温度是86°F,以摄氏度计算是多少?

9. If a $1,200 investment earns $168 over 4 years, what is the annual interest rate?
::9. 如果1 200美元投资在4年内赚取168美元,年利率是多少?

For questions 10-11, use the equation for the surface area of a cylinder, $\begin{aligned} S = 2 π r^{2} + 2 π r h \end{aligned}$ , where $\begin{aligned} r \end{aligned}$ is the radius and $\begin{aligned} h \end{aligned}$ is the height.
::对于问题10-11,使用圆柱体表面的方程式S=2r2+2rh,圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆的圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆圆

10. Solve the equation for $\begin{aligned} h \end{aligned}$ .
::10. 解决h的等式

11. Find $\begin{aligned} h \end{aligned}$ if the surface area is $\begin{aligned} 120 π {cm}^{2} \end{aligned}$ and the radius is 6 cm.
::11. 查找h,如果表面面积为120厘米2,半径为6厘米。

Explore More
::探索更多

1. In the next chapter, we are going to cover linear equations in two variables, that is, equations of the form $\begin{aligned} y = m x + b \end{aligned}$ . This is called slope-intercept form. Equations in the form $\begin{aligned} A x + B y = C \end{aligned}$ are said to be in general form.
::1. 在下一章中,我们将以两个变量,即y=mx+b的方程式,涵盖线性方程。这称为斜度干扰方程。Ax+By=C的方程式据说是一般性的。

Solve for $\begin{aligned} y \end{aligned}$ in the equation $\begin{aligned} \frac{3 x}{2} - \frac{4 y}{5} = 12 \end{aligned}$ and thus convert this equation from general form into slope-intercept form.
::在 3x2 - 4y5 = 12 方程式中为 y 解决 y , 从而将此方程式从一般形式转换为斜坡界面形式。

by CK-12 shows how to solve a similar problem.
::的 CK-12 显示如何解决类似的问题。

2. Recall that a prism is a shape that has the same cross-section throughout the entire shape. For example, a cylinder is a prism because its cross-section is always a circle and a box. A rectangular prism is also a prism because its cross-section is always a rectangle.
::2. 回顾棱晶是一种形状,在整个形状中具有相同的横截面,例如,圆柱体是一个棱晶,因为它的横截面始终是一个圆形和一个盒子,矩形棱晶也是棱晶,因为它的横截面始终是一个矩形。

To find the volume of a prism, you can multiply the area of the cross-section by the height.
::要找到棱镜的体积,您可以将横截面的面积乘以高度。

by Valerie Ratcliff demonstrates how to find the volume of 3D shapes.
::瓦莱丽·拉特克利夫展示了如何找到 3D 形状的体积。

Consider a cylinder and a rectangular prism with a square cross-section that have the same volume. If the diameter of the cylinder and the side of the square are both 10 cm, which is taller?
::考虑一个圆柱体和一个具有平方横截面的长方形棱柱,其体积相同。如果圆柱体的直径和平方的侧面都为10厘米,则高多少?

3. Patsy is trying to determine how much wallpaper she needs to cover one rectangular wall in her dining room. She originally planned to only put wallpaper half-way up the wall and then paint the top half of the wall. However, she changed her mind and now wants to wallpaper the entire wall. How does doubling the height impact the total area of the space she will be wallpapering?
::3. Patsy正试图确定她需要多少壁纸来遮盖她餐厅的一面长方形墙,她最初只打算把墙纸放在墙上半边,然后漆墙的顶部,然而,她改变了主意,现在想把整个墙盖上墙。

4. You are helping create a circular flower garden at a senior center. You have flowers needed to line the circumference of 31.4 feet. You want to plant flowers across the diameter. Calculate the diameter of the flower bed.
::4. 您正在帮助在高级中心建立一个圆形花园。您需要花朵来排列31.4英尺的周长。您想要在直径上种植花朵。计算花床的直径。

5. You are renting a moving truck and need to find one with enough space to carry your furniture. All the trucks have dimensions of 7 feet tall by 6 feet wide, but the length varies. You need a truck with a volume of 630 cubic feet. What is the length of the truck you need?
::5. 您正在租用一辆流动卡车,需要找到一辆有足够空间运载家具的卡车,所有卡车都有7英尺高6英尺宽的尺寸,但长度不同,需要一辆630立方英尺的卡车。您需要的卡车长度是多少?

Answers to Review and Explore More Problems
::对审查和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces a concept explored in this section:
::试一试,这一互动强化了本节所探讨的概念:

Section outline

Solving Literal Equations ::解决文学等量

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 to Review and Explore More Problems ::对审查和探讨更多问题的答复

PLIX ::PLIX