课程： 1.6 变量的溶解代数等量

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变量的溶解代数等量

You are planning a trip to Spain in the summer. In the U.S., we use Fahrenheit to measure the temperature, but in Europe, they use Celsius. To get yourself used to Celsius, you find the conversion equation $\begin{align*}F=\frac{9}{5}C+32\end{align*}$ . You can substitute the temperature in Celsius for C and solve the equation for the temperature in Fahrenheit, F . When you start looking at the weather in Spain in the summer, the website says it is usually $\begin{align*}35^\circ C\end{align*}$ . What will the temperature be in Fahrenheit?
::你计划夏天去西班牙旅行。在美国,我们用法赫伦海特测量温度,但在欧洲,他们使用摄氏度。为了让自己习惯于摄氏度,你能找到转换方程式F=95C+32。你可以用摄氏度代替C,并用法赫伦海特的温度解析方程式。当你开始观察夏季西班牙的天气时,网站说通常为35°C。法赫伦海特的温度是多少?

Solving for a Variable in an Algebraic Equation
::代数等量变量的溶解

Solving an algebraic equation for a variable can be tricky, but it can also be very useful. This technique can be used to go back and forth between two different units of measurement. To solve for, to isolate , a variable within an equation, you must undo the operations that are in the equation.
::解决变量的代数方程式可能比较棘手, 但也可能非常有用。此技术可以用于在两个不同的计量单位之间往返。要解决, 要分离一个方程式中的变量, 您必须撤销公式中的操作。

Let's solve each equation for the specified variable.
::让我们为指定的变量解析每个方程式。

Solve for $\begin{align*}x\end{align*}$ in the equation $\begin{align*}3x-4y=12\end{align*}$ .
::在 3x--4y=12 方程式中解决 x 的 x 值。

To solve for $\begin{align*}x\end{align*}$ , you need to move the $\begin{align*}4y\end{align*}$ to the other side of the equation. In order to do this, we must do the opposite operation . Since the $\begin{align*}4y\end{align*}$ is being subtracted, we must add it to the other side of the equation.
::要解决 x, 您需要将 4y 移动到等式的另一面。为了做到这一点, 我们必须执行相反的操作。由于 4y 正在被减去, 我们必须将其添加到等式的另一面。

\begin{aligned} 3 x - 4 y = 12 \\ \underline{+ 4 y = + 4 y} \\ 3 x = 4 y + 12 \end{aligned}

$\begin{align*}& \ 3x-4y=12\\ & \underline{ \quad \ \ +4y=+4y \; \; \; \; \;}\\ & \qquad \ 3x=4y+12\end{align*}$
::3x-4y=12+4y4y_ 3x=4y+12

Now we need to get $\begin{align*}x\end{align*}$ alone. $\begin{align*}3x\end{align*}$ is the same as “3 multiplied by $\begin{align*}x\end{align*}$ .” Therefore , to undo the multiplication we must divide both sides of the equation by 3.
::现在我们需要单独获得 x。 3x与“ 3乘x ” 相同。因此,要取消乘法,我们必须将方程两侧除以3。

\begin{aligned} \frac{3 x}{3} & = \frac{4 y}{3} + \frac{12}{3} \\ x & = \frac{4}{3} y + 4 \end{aligned}

$\begin{align*}\frac{3x}{3} &= \frac{4y}{3}+\frac{12}{3}\\ x &= \frac{4}{3}y+4\end{align*}$
::3x3=4y3+123x=43y+4

When undoing multiplication, you must divide everything in the equation by 3.
::当取消乘法时,您必须将方程式中的一切除以 3 。

A few things to note:
::值得一提的有几件事:

To undo an operation within an equation, do the opposite.
::要在等式内撤销操作, 就得反其道而行之。

Perform the opposite operations in the reverse order of the .
::执行反向操作。

Combine all like terms on the same side of the equals sign before doing the first bullet.
::在做第一颗子弹之前,将相同条件的同一边加在一起。

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

First, isolate $\begin{align*}b\end{align*}$ . Move both the $\begin{align*}6a\end{align*}$ and the 4 over to the other side by doing the opposite operation.
::首先,隔离b. 将6a和4两边移到另一边,进行相反的行动。

\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}

$\begin{align*}& \frac{2b}{3}+6a-4=8\\ & \underline{\quad \ \ -6a+4=+4-6a}\\ & \qquad \qquad \frac{2b}{3}=12-6a\end{align*}$
::2b3+6a-4=8-6a+44-6a_2b3=12-6a

Now, we have a fraction multiplied by $\begin{align*}b\end{align*}$ . To undo this we must multiply by the reciprocal of $\begin{align*}\frac{2}{3}\end{align*}$ , which is $\begin{align*}\frac{3}{2}\end{align*}$ . This means, 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。这意味着,我们必须将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 b}{3} = \frac{3}{2} \cdot 12 - \frac{3}{2} \cdot 6 a \\ b = 18 - 9 a \end{aligned}

$\begin{align*}& \quad \ \frac{2b}{3} = 12-6a\\ & \frac{3}{2} \left( \frac{2b}{3} = 12-6a \right)\\ & \frac{3}{2} \cdot \frac{2b}{3} = \frac{3}{2} \cdot 12-\frac{3}{2} \cdot 6a\\ & \qquad b = 18-9a\end{align*}$
::2b3=12-6a32(2b3=12-6a) 322&2b3=3212-32-326ab=18-9a

Even though we know that a 9 can be pulled out of the 18 and the 9 in the above equation, it is not a necessary step in solving for $\begin{align*}b\end{align*}$ . Now, we can do what the question asks, find $\begin{align*}b\end{align*}$ when $\begin{align*}a = 1\end{align*}$ and -2. Plug in these values for $\begin{align*}a\end{align*}$ .
::尽管我们知道可以从上述等式中的18和9中抽出9,但这不是解决b的必要步骤。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 : 18 - 9 (- 2) = 18 + 18 = 36 \end{aligned}

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

The area of a triangle is $\begin{align*}A=\frac{1}{2}bh\end{align*}$ , where $\begin{align*}b\end{align*}$ is the base of the triangle and $\begin{align*}h\end{align*}$ is the height. You know that the area of a triangle is $\begin{align*}60 \ in^2\end{align*}$ 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。

First solve the equation for $\begin{align*}h\end{align*}$ .
::先解答h的方程。

\begin{aligned} A & = \frac{1}{2} b h \\ 2 \cdot A & = 2 \cdot \frac{1}{2} b h & Multiply both sides by 2. \\ 2 A & = b h & Divide both sides by b . \\ \frac{2 A}{b} & = h \end{aligned}

$\begin{align*}A &= \frac{1}{2}bh\\ 2 \cdot A &= 2 \cdot \frac{1}{2}bh && \text{Multiply both sides by 2.}\\ 2A &= bh && \text{Divide both sides by} \ b.\\ \frac{2A}{b} &= h\end{align*}$
::A=12bh2}A=212bh 以2.2A=bhDivide 以b.2Ab=h 表示两边的 A=12bh2}A=212bh 以2.2A=bhDivide 以b.2Ab=h

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

This process is helpful when solving for any variable in any equation or formula . Some formulas you may need to know for this text are listed below.
::在解决任何方程式或公式中的任何变量时,此进程很有帮助。以下列出了您可能需要知道的用于此文本的某些公式。

Distance	$\begin{align}d = rt\end{align}$	$\begin{align}d=\text{distance}, r = \text{rate}, t = \text{time}\end{align}$
Temperature	$\begin{align}F= \frac{9}{5}C+32\end{align}$	$\begin{align}F=\text{degrees in Fahrenheit}, C = \text{degrees in Celsius}\end{align}$
Area of a Triangle	$\begin{align}A = \frac{1}{2}bh\end{align}$	$\begin{align}A=\text{area}, b = \text{base}, h = \text{height}\end{align}$
Area of a Rectangle	$\begin{align}A=bh\end{align}$	$\begin{align}A = \text{area}, b = \text{base}, h = \text{height}\end{align}$
Area of a Circle	$\begin{align}A=\pi r^2\end{align}$	$\begin{align}A = \text{area}, r = \text{radius}\end{align}$
Area of a Trapezoid	$\begin{align}A=\frac{1}{2}h(b_1+b_2)\end{align}$	$\begin{align}A = \text{area}, h = \text{height}, b_1 = \text{one base}, b_2 = \text{other base}\end{align}$
Perimeter of a Rectangle	$\begin{align}P=2l+2w\end{align}$	$\begin{align}P = \text{perimeter}, l = \text{length}, w = \text{width}\end{align}$
Circumference of a Circle	$\begin{align}C=2 \pi r\end{align}$	$\begin{align}C = \text{circumference}, r = \text{radius}\end{align}$

Examples
::实例

Example 1
::例1

Earlier, you were asked to convert Celsius into Fahrenheit.
::早些时候,你被要求将摄氏转换成华氏。

First, let's substitute in $\begin{align*}C=35^\circ\end{align*}$ to find the degrees in Fahrenheit.
::首先,我们用C=35来取代法氏华氏度。

\begin{aligned} F & = \frac{9}{5} \cdot 35 + 32 \\ F & = 63 + 32 \\ F & = 95^{\circ} \end{aligned}

$\begin{align*}F&=\frac{9}{5}\cdot 35 + 32\\ F&=63 + 32 \\ F&=95^\circ\end{align*}$
::F=95-35+32F=63+32F=95

As you can see, a good estimator for the conversion is to double the degrees, when in Celsius, and then add 32. Needless to say, it is going to be quite hot in Spain and you should pack summer clothes for your trip.
::如你所见, 转换的良好估计值是摄氏度的两倍, 然后加上32。不用说, 西班牙的温度会相当高,

Example 2
::例2

Solve $\begin{align*}\frac{3}{4}x-2y=15\end{align*}$ for $\begin{align*}x\end{align*}$ .
::x 的溶解 34x-2y=15 。

First add $\begin{align*}2y\end{align*}$ to both sides, then multiply both sides by the reciprocal of $\begin{align*}\tfrac{3}{4}\end{align*}$ .
::首先向双方增加2y,然后以34对等乘以双方。

\begin{aligned} \frac{3}{4} x - 2 y = 15 \\ \underline{+ 2 y + 2 y} \\ \frac{3}{4} x = 2 y + 15 \\ x = \frac{4}{3} \cdot 2 y + \frac{4}{3} \cdot 15 \\ x = \frac{8}{3} y + 20 \end{aligned}

$\begin{align*}& \ \frac{3}{4}x-2y=15\\ & \underline{\quad \ \ +2y \ +2y \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;}\\ & \qquad \frac{3}{4}x=2y+15\\ & \qquad \ \ x=\frac{4}{3} \cdot 2y+\frac{4}{3} \cdot 15\\ & \qquad \ \ x=\frac{8}{3}y+20\end{align*}$
::34 - 2y=15+2y +2y_ 34x=2y+15 x=43=2y+43=15 x=83y+20

Example 3
::例3

If the temperature is $\begin{align*}41^\circ F\end{align*}$ , what is it in Celsius?
::如果温度是41°F 摄氏度是多少?

You can solve this problem two different ways: First, plug in 41 for $\begin{align*}F\end{align*}$ and then solve for $\begin{align*}C\end{align*}$ or solve for $\begin{align*}C\end{align*}$ first and then plug in 41 for $\begin{align*}F\end{align*}$ . We will do the second option.
::您可以以两种不同的方式解决这个问题:首先,F在41中插入,然后C在41中解决,C在1中解决,F在41中插入。我们将做第二个选择。

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

$\begin{align*}F &= \frac{9}{5}C+32\\ F-32 &= \frac{9}{5}C\\ \frac{5(F-32)}{9} &= C\end{align*}$
::F=95C+32F-32=95C5(F-32)9=C

Now, plug in 41 for $\begin{align*}F\end{align*}$ .
::现在,插入41F。

\begin{aligned} C & = \frac{5 (41 - 32)}{9} \\ C & = \frac{5 \cdot 9}{9} \\ C & = 5^{\circ} \end{aligned}

$\begin{align*}C&=\frac{5(41-32)}{9}\\C&=\frac{5 \cdot 9}{9}\\C&=5^\circ \end{align*}$
::C=5(41-32)9C=599C=5

Review
::回顾

Solve the following equations or formulas for the indicated variable.
::为指定变量解决下列方程式或公式。

$\begin{align*}6x-3y=9\end{align*}$ ; solve for $\begin{align*}y\end{align*}$ .
::6x-3y=9; y 的解决方案。

$\begin{align*}4c+9d=16\end{align*}$ ; solve for $\begin{align*}c\end{align*}$ .
::4c+9d=16; 解析 c.

$\begin{align*}5f-6g=14\end{align*}$ ; solve for $\begin{align*}f\end{align*}$ .
::5f-6g=14; f.

$\begin{align*}\frac{1}{3}x+5y=1\end{align*}$ ; solve for $\begin{align*}x\end{align*}$ .
::13x+5y=1; x 的解析。

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

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

$\begin{align*}P=2l+2w\end{align*}$ ; solve for $\begin{align*}w\end{align*}$ .
::P=2l+2w; 解决 w.

$\begin{align*}F=\frac{9}{5}C+32\end{align*}$ ; solve for $\begin{align*}C\end{align*}$ .
::F=95C+32; C的解析。

Find the value of $\begin{align*}y\end{align*}$ given the value of $\begin{align*}x\end{align*}$ .
::查找 y 值,给 x 值。

$\begin{align*}4x-8y=2; x = -1\end{align*}$
::4 - 8y=2;x% 1

$\begin{align*}2y-5x=12; x = 16\end{align*}$
::2y-5x=12;x=16

$\begin{align*}3x+\frac{1}{2}y=-5; x = 7\end{align*}$
::3x+12y5;x=7

$\begin{align*}\frac{1}{4}x+\frac{2}{3}y-18=0; x = -24\end{align*}$
::14x+23y- 18=0;x24

For questions 13-15, use the formulas in the chart above to answer the following questions.
::对于问题13-15,使用上图中的公式回答下列问题。

If the temperature is $\begin{align*}86^\circ F\end{align*}$ , what is it in Celsius?
::如果温度是86°F, 以摄氏度算什么?

If the area of a circle is $\begin{align*}36 \pi \ cm^2\end{align*}$ , what is the radius?
::如果圆的面积是36-cm2,圆的半径是多少?

The area of a trapezoid is $\begin{align*}72 \ ft^2\end{align*}$ , the height is 8 ft and $\begin{align*}b_1\end{align*}$ is 6, what is the length of the other base?
::类区域为72平方英尺,高度为8平方英尺,b1是6,其他基数的长度是多少?

For questions 16-17, use the equation for the surface area of a cylinder, $\begin{align*}SA=2 \pi r^2+2 \pi rh\end{align*}$ , where $\begin{align*}r\end{align*}$ is the radius and $\begin{align*}h\end{align*}$ is the height.
::对于问题16-17,使用圆柱体表面的方程式SA=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

Solve the equation for $\begin{align*}h\end{align*}$ .
::解决h的方程

Find $\begin{align*}h\end{align*}$ if the surface area is $\begin{align*}120\pi \ cm^2\end{align*}$ and the radius is 6 cm.
::h 查找表面面积是否为120厘米2,半径为6厘米。

Challenge The formula for the volume of a sphere is $\begin{align*}V=\frac{4}{3}\pi r^3\end{align*}$ . Solve the equation for $\begin{align*}r\end{align*}$ , the radius.
::挑战球体体积的公式是 V=43°r3. 解析 r, 半径的方程。

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。

Distance	$\begin{align}d = rt\end{align}$	$\begin{align}d=\text{distance}, r = \text{rate}, t = \text{time}\end{align}$
Temperature	$\begin{align}F= \frac{9}{5}C+32\end{align}$	$\begin{align}F=\text{degrees in Fahrenheit}, C = \text{degrees in Celsius}\end{align}$
Area of a Triangle	$\begin{align}A = \frac{1}{2}bh\end{align}$	$\begin{align}A=\text{area}, b = \text{base}, h = \text{height}\end{align}$
Area of a Rectangle	$\begin{align}A=bh\end{align}$	$\begin{align}A = \text{area}, b = \text{base}, h = \text{height}\end{align}$
Area of a Circle	$\begin{align}A=\pi r^2\end{align}$	$\begin{align}A = \text{area}, r = \text{radius}\end{align}$
Area of a Trapezoid	$\begin{align}A=\frac{1}{2}h(b_1+b_2)\end{align}$	$\begin{align}A = \text{area}, h = \text{height}, b_1 = \text{one base}, b_2 = \text{other base}\end{align}$
Perimeter of a Rectangle	$\begin{align}P=2l+2w\end{align}$	$\begin{align}P = \text{perimeter}, l = \text{length}, w = \text{width}\end{align}$
Circumference of a Circle	$\begin{align}C=2 \pi r\end{align}$	$\begin{align}C = \text{circumference}, r = \text{radius}\end{align}$

章节大纲

常规

变量的溶解代数等量

Solving for a Variable in an Algebraic Equation ::代数等量变量的溶解

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)