Section: 平等两步平等和平等财产 | 7.12 平等的两个层次的平等和平等财产

Section outline

As a mid- term project in a music class, some students have the opportunity to see a live jazz concert and describe the music to the class. The music teacher, Mr. Cooper, purchased some tickets. The service fee to buy tickets to the jazz concert is 9 dollars, and each ticket costs 12 dollars. If the total cost of the tickets was 93 dollars, can you figure out how many tickets Mr. Cooper his bought?
::作为音乐班的中期项目,一些学生有机会观看一次爵士音乐现场音乐会,并向课堂描述音乐。音乐老师库珀先生购买了一些票。购买爵士音乐会票的服务费为9美元,每张票费用为12美元。如果机票总费用为93美元,你能弄清楚库珀先生购买了多少票吗?

In this concept, you will learn to solve two-step equations.
::在这个概念中,你会学会解决两步方程。

Solving Two-Step Equations
::解决双层平方

Sometimes, when the numbers are small integers , you might be able to solve a two-step equation by thinking about it. For instance, can you solve $\begin{aligned} 3 x + 3 = 9 \end{aligned}$ ?
::有时,当数字是小整数时, 你也许能够通过思考来解析一个两步方程。例如, 您可以解析 3x+3=9 吗 ?

Here the numbers are small. You can probably look at this equation and ask yourself, “What number times three plus three is nine?” The logical answer is 2. You can check your answer by substituting 2 in for $\begin{aligned} x \end{aligned}$ , to see if both sides of the equation are the same. If they are, then your work is accurate.
::这里的数字是小的。您可能可以看到这个方程, 并问自己, “ 多少乘以三加三是九? ” 逻辑答案是 2 。您可以用 X 替换 2 来检查答案, 查看方程的两边是否相同。如果是, 那么您的工作是准确的。

Let’s plug in 2 for $\begin{aligned} x \end{aligned}$ to check that possible answer.
::让我们在 2 中插入 x 来检查可能的答案。

$\begin{aligned} \begin{array}{rcl} 3 (2) + 3 & = & 9 \\ 6 + 3 & = & 9 \\ 9 & = & 9 \end{array} \end{aligned}$

Since it is true that $\begin{aligned} 9 = 9 \end{aligned}$ , the answer $\begin{aligned} x = 2 \end{aligned}$ works.
::由于9=9确实有效,回答 x=2 有效。

As equations get more complex it is important to use properties of equality to isolate the variable and solve the equation.
::随着方程式变得更加复杂,必须使用平等的属性来孤立变量并解决方程式问题。

Here are the properties of equality you need to isolate terms and solve equations.
::这是平等的特点您需要分离条件和解析方程式。

The Subtraction Property of Equality is used when you have an equation with addition in it. It states that you can subtract the same quantity from both sides of the equation without changing the equality.
::平等减法属性当您在等式上加上附加等式时使用,它指出,您可以在不改变平等的情况下从等式两侧减去相同数量。

The Addition Property of Equality is used when you have an equation with subtraction in it. It states that you can add the same quantity to both sides of the equation without changing the equality.
::平等附加财产等式等式用等式用等式用减法 , 规定等式的双方均可以增加相同的数量 , 而不改变等式等式的等式。

The Division Property of Equality is used when you have an equation with a variable multiplied by a number. It states that you can divide both sides of an equation by the same quantity (as long as that quantity is not equal to zero) without changing the equality.
::“平等财产”是当您用变量乘以一个数字的方程式时使用的。它指出,您可以在不改变平等的情况下,用相同数量(只要数量不等于零)将方程式的两边除以相同数量(只要数量不等于零)。

The Multiplication Property of Equality is used when you have an equation with a variable divided by a number. It states that you can multiply both sides of an equation by the same quantity without changing the equality.
::“平等乘法属性”在您使用变量除以数字的方程式时使用。它指出,您可以在不改变平等的情况下,以相同数量乘以方程式的两侧。

Let’s look at an example and use properties of equality to isolate the variable and solve the equation.
::让我们举个例子, 使用平等的属性来孤立变量并解答等式。

$\begin{aligned} 2 + 3 n = 11 \end{aligned}$

::2+3n=11

There are two terms on the left side of the equation, $\begin{aligned} 2 \end{aligned}$ and $\begin{aligned} 3 n \end{aligned}$ .
::方程左侧有两个条件, 2和3n。

The first step is to get the term with the variable, $\begin{aligned} 3 n \end{aligned}$ , by itself on one side of the equal $\begin{aligned} (=) \end{aligned}$ sign.
::第一步是以变量 3n 获得该术语, 单靠等于 (=) 符号的一面。

In the equation, 2 is added to $\begin{aligned} 3 n \end{aligned}$ . So, you use the inverse of addition, which is subtraction, and subtract 2 from both sides of the equation.
::在等式中, 2 添加到 3n 。所以, 您使用反向加法, 即减法, 然后从方程式的两侧减去 2 。

You can do this because of the subtraction property of equality. That property states that in order to keep the values on both sides of the equation equal, whatever is subtracted from one side of the equation must also be subtracted from the other side.
::您可以这样做, 原因是平等减值财产。该财产声明, 为了保持方程两侧的数值相等, 无论从方程的一边减去什么, 也必须从另一方扣除。

Let’s see what happens when we subtract 2 from both sides of the equation
::让我们看看当我们从等式的两面减去2时会发生什么。

$\begin{aligned} \begin{array}{rcl} 2 + 3 n & = & 11 \\ 2 - 2 + 3 n & = & 11 - 2 \\ 0 + 3 n & = & 9 \\ 3 n & = & 9 \end{array} \end{aligned}$

::2+3n=112-2+3n=11-20+3n=9

Now the problem is much more simple. You have reduced a two-step equation to a one-step equation .
::现在问题很简单。你把两步方程减为一步方程。

Next, use the division property of equality and divide both sides of the equation by 3. Then simplify.
::其次,使用平等分割财产,将等式的两侧除以3,然后简化。

$\begin{aligned} \begin{array}{rcl} 3 n & = & 9 \\ \frac{3 n}{3} & = & \frac{9}{3} \\ 1 n & = & 3 \\ n & = & 3 \end{array} \end{aligned}$

::3n=93n3=931n=3n=3n=3

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

Here is another example.
::下面是另一个例子。

Solve $\begin{aligned} 2 x - 5 = 11 \end{aligned}$ for $\begin{aligned} x \end{aligned}$ .
::x 的溶解 2x-5=11 。

This is a two-step equation. The ultimate goal is to isolate the variable $\begin{aligned} x \end{aligned}$ . This will be easier to do if term with the variable, $\begin{aligned} 2 x \end{aligned}$ is by itself on one side of the equation.
::这是一个两步方程。最终目标是将变量 x 分离出来。如果使用变量, 2x 本身在方程的一边, 这样做比较容易。

First, to get $\begin{aligned} 2 x \end{aligned}$ by itself on one side of the equation, use the addition property of equality to add 5 to both sides of the equation.
::首先,为了在等式的一面获得2x, 使用平等的附加属性在等式的两面加5。

$\begin{aligned} \begin{array}{rcl} 2 x - 5 & = & 11 \\ 2 x - 5 + 5 & = & 11 + 5 \\ 2 x + 0 & = & 16 \\ 2 x & = & 16 \end{array} \end{aligned}$

::2x-5=112x-5+5=11+52x+0162x=16

Now, the two-step equation is a one-step equation and is much easier to solve.
::现在,两步方程是一个一步方程,很容易解决。

Next, use the division property of equality, and divide both sides of the equation by 2, to isolate the variable $\begin{aligned} x \end{aligned}$ .
::其次,使用平等分割属性,将等式两侧除以2,以孤立变量x。

$\begin{aligned} \begin{array}{rcl} \frac{2 x}{2} & = & \frac{16}{2} \\ 1 x & = & 8 \\ x & = & 8 \end{array} \end{aligned}$
The answer is $\begin{aligned} x = 8 \end{aligned}$ .
::2x2=1621x=8x=8 答案是 x=8。

Here is another example.
::下面是另一个例子。

Solve $\begin{aligned} \frac{x}{5} - 8 = 17 \end{aligned}$ for $\begin{aligned} x \end{aligned}$ .
::x 的溶解 x5 - 8=17 。

First, use the addition property of equality to get $\begin{aligned} \frac{x}{5} \end{aligned}$ by itself on one side of the equation. Add 8 to both sides of the equation.
::首先,使用增加的平等属性来获得 X5 本身在方程的一边。在方程的两侧加8。

$\begin{aligned} \begin{array}{rcl} \frac{x}{5} - 8 & = & 17 \\ \frac{x}{5} + - 8 + 8 & = & 17 + 8 \\ \frac{x}{5} + 0 & = & 25 \\ \frac{x}{5} & = & 25 \end{array} \end{aligned}$

::x5-8=17x58+8=17+8x5+0=25x5=25

Now, the two-step equation has been reduced to a one-step equation. Since $\begin{aligned} x \end{aligned}$ is divided by 5, you need to use the inverse of division , multiplication , to isolate the variable $\begin{aligned} x \end{aligned}$ .
::现在, 两步方程已降为一步方程。由于 x 被除以 5, 您需要使用正反除法乘法来分隔变量 x 。

Next, use the multiplication property of equality, and multiply both sides of the equation by 5. Then simplify both sides of the equation.
::其次,使用平等的乘法属性,将等式的两侧乘以5。然后简化等式的两侧。

$\begin{aligned} \begin{array}{rcl} \frac{x}{5} \times 5 & = & 25 \times 5 \\ x \times \frac{5}{5} & = & 125 \\ x \times 1 & = & 125 \\ x & = & 125 \end{array} \end{aligned}$

::x5x5=25x5xxxxx55=125x1x1=125x=125

The answer is $\begin{aligned} x = 125 \end{aligned}$ .
::答案是 x=125。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Mr. Cooper and his music class.
::早些时候,你被问及 Cooper 先生和他的音乐课

He bought tickets to an amazing jazz concert for some students. There was a $9 service fee per order, and each ticket cost $12. The total cost for the order was $93. Can you write an equation to represent this situation and then solve it?
::他为一些学生购买了很棒的爵士音乐会的票。每张票有9美元的服务费,每张票12美元。订单的总费用是93美元,你能写一个方程式来代表这种情况然后解决吗?

First, let $\begin{aligned} n \end{aligned}$ be the number of tickets bought. The total cost is 9 plus 12 times the number of tickets bought. This total cost is 93. Translate this into an equation.
::首先,请看买票的数量。总成本是买票数量的9+12倍。总成本是93。把它转换成一个方程式。

$\begin{aligned} 12 n + 9 = 93 \end{aligned}$

::12n+9=93

The answer is $\begin{aligned} 12 n + 9 = 93 \end{aligned}$ .
::答案是12n+9=93

Next, solve the two-step equation.
::接下来,解决两步方程

First isolate $\begin{aligned} 12 n \end{aligned}$ . Use the subtraction property of equality and subtract 9 from both sides of the equation.
::首先分离 12n. 使用等式的减法属性,从等式的两侧减去9。

$\begin{aligned} \begin{array}{rcl} 12 n + 9 & = & 93 \\ 12 n + 9 - 9 & = & 93 - 9 \\ 12 n + 0 & = & 84 \\ 12 n & = & 84 \end{array} \end{aligned}$

::12n+9=9312n+9-9=93-912n+0=8412n=84

Now, solve the one-step equation. Use the division property of equality and divide both sides of the equation by 12.
::现在,解决单步方程。使用平等分割属性,将方程的两边除以12。

$\begin{aligned} \begin{array}{rcl} \frac{12 n}{12} & = & \frac{84}{12} \\ \frac{12}{12} n & = & 7 \\ 1 n & = & 7 \\ n & = & 7 \end{array} \end{aligned}$

::12n12=84122112n=71n=7n=7n=7

The answer is Mr. Cooper bought 7 tickets.
::答案是库珀先生买了七张票

Example 2
::例2

A landscaper charges $35 for each landscaping job, plus $20 for each hour worked. After one landscaping job, the landscaper charged $95.
::园艺人员为每个园艺工作收费35美元,每工作小时加20美元。在一次园艺工作之后,园艺人员收费95美元。

Write an algebraic equation to represent $\begin{aligned} h \end{aligned}$ , the number of hours that the landscaper worked on that $95 job.
::写一个代数方程式来代表h, 景观设计师在那95美元的作业上工作的时间。

How many hours did that job take?
::这份工作花了多少小时?

Consider part a first.
::考虑第一部分。

The landscaper earned $20 for each hour worked on that job, so you multiply $20 by $\begin{aligned} h \end{aligned}$ , the number of hours worked, to find how the landscaper charged for the hours worked, and then add the initial $35. This equals the total charge of $95.
::园艺人员每工作一个小时挣20美元,所以你乘以20乘以小时,即工作小时数,寻找园艺人员如何为工作小时收取费用,然后加上最初的35美元。这等于总收费95美元。

The answer is $\begin{aligned} 20 h + 35 = 95 \end{aligned}$ .
::答案是20小时+35=95

Next, consider part b .
::下一步,考虑B部分。

Solve the equation to find the number of hours the landscaper worked on that job.
::解决这个方程以找出园艺师在那份工作上工作的时间

First, to isolate the term $\begin{aligned} 20 h \end{aligned}$ use the subtraction property of equality and subtract $\begin{aligned} 35 \end{aligned}$ from both sides of the equation.
::首先,将20h一词分开使用等值减法,从等式的两侧减去35。

$\begin{aligned} \begin{array}{rcl} 20 h + 35 - 35 & = & 95 - 35 \\ 20 h + 0 & = & 60 \\ 20 h & = & 60 \end{array} \end{aligned}$

::20小时+35-35=95-3520小时+0=6020小时=60

The two-step equation has been reduced to a one-step equation.
::两步方程已减为一步方程。

Next, use the division property of equality to isolate the variable $\begin{aligned} h \end{aligned}$ , and divide both sides of the equation by 20.
::其次,使用平等分割属性来孤立变数h,并将等式的两边除以20。

$\begin{aligned} \begin{array}{rcl} \frac{20 h}{20} & = & \frac{60}{20} \\ \frac{20}{20} h & = & 3 \\ 1 h & = & 3 \\ h & = & 3 \end{array} \end{aligned}$

::2020=6020202020h=31h=3h=3

The answer is the landscaper worked for 3 hours on the $95 job.
::答案是景观设计师在95美元的工作上工作了3小时。

Solve each equation.
::解决每个方程式

Example 3
::例3

$\begin{aligned} 5 x + 7 = 32 \end{aligned}$

::5x+7=32

First, use the subtraction property of equality and subtract 7 from both sides of the equation.
::首先,使用等式的减法财产,减去等式两侧的7。

$\begin{aligned} \begin{array}{rcl} 5 x + 7 - 7 & = & 32 - 7 \\ 5 x + 0 & = & 25 \\ 5 x & = & 25 \end{array} \end{aligned}$

::5x+7-7=32-75x+0=2555x=25

Next, use the division property of equality and divide each side of the equation by 5.
::其次,使用平等分割财产,将等式的每侧除以5。

$\begin{aligned} \begin{array}{rcl} \frac{5 x}{5} & = & \frac{25}{5} \\ \frac{5}{5} x & = & 5 \\ 1 x & = & 5 \\ x & = & 5 \end{array} \end{aligned}$
The answer is $\begin{aligned} x = 5 \end{aligned}$ .
::5x5=255555x=51x=5x=5 答案是 x=5。

Example 4
::例4

$\begin{aligned} 3 a + 9 = 39 \end{aligned}$

::3a+9=39

First, use the subtraction property of equality and subtract 9 from both sides of the equation.
::首先,使用等式的减法财产,减去等式两侧的9。

$\begin{aligned} \begin{array}{rcl} 3 a + 9 - 9 & = & 39 - 9 \\ 3 a + 0 & = & 30 \\ 3 a & = & 30 \end{array} \end{aligned}$

::3a+9-9=39-93a+0=303a=30

Next, use the division property of equality and divide both sides of the equation by 3.
::其次,使用平等分割财产,将等式两侧除以3。

$\begin{aligned} \begin{array}{rcl} \frac{3 a}{3} & = & \frac{30}{3} \\ \frac{3}{3} a & = & 10 \\ 1 a & = & 10 \\ a & = & 10 \end{array} \end{aligned}$
The answer is $\begin{aligned} a = 10 \end{aligned}$ .
::3a3=30333a=101a=10a=10a=10 答案是a=10。

Example 5
::例5

$\begin{aligned} \frac{y}{4} - 8 = 4 \end{aligned}$

::y4 - 8=4 y4 - 8=4

First, use the addition property of equality and add 8 to both sides of the equation.
::首先,增加平等财产,在等式的两侧增加8个。

$\begin{aligned} \begin{array}{rcl} \frac{y}{4} - 8 + 8 & = & 4 + 8 \\ \frac{y}{4} + 0 & = & 12 \\ \frac{y}{4} & = & 12 \end{array} \end{aligned}$

::y4-8+8=4+8y4+0=12y4=12

Next, use the multiplication property of equality and multiply both sides of the equation by 4.
::其次,使用平等的乘法属性,将等式的两侧乘以4。

$\begin{aligned} \begin{array}{rcl} \frac{y}{4} \times 4 & = & 12 \times 4 \\ y \times \frac{4}{4} & = & 48 \\ y \times 1 & = & 48 \\ y & = & 48 \end{array} \end{aligned}$

::y4x4=12x4yx44=48yx1=48y=48

The answer is $\begin{aligned} y = 48 \end{aligned}$ .
::答案是y=48

Review
::回顾

Solve each two-step equation for the unknown variable.
::为未知变量解决每个两步方程。

$\begin{aligned} 3 x + 2 = 14 \end{aligned}$
::3x+2=14

$\begin{aligned} 6 y + 5 = 29 \end{aligned}$
::6y+5=29

$\begin{aligned} 7 x + 3 = 24 \end{aligned}$
::7x+3=24

$\begin{aligned} 5 x + 7 = 42 \end{aligned}$
::5x+7=42

$\begin{aligned} 6 y + 1 = 43 \end{aligned}$
::6y+1=43

$\begin{aligned} 9 a + 7 = 88 \end{aligned}$
::9a+7=88

$\begin{aligned} 11 b + 12 = 56 \end{aligned}$
::11b+12=56

$\begin{aligned} 12 x - 3 = 21 \end{aligned}$
::12 - 3=21

$\begin{aligned} 4 y - 5 = 19 \end{aligned}$
::4y-5=19

$\begin{aligned} 3 a - 9 = 21 \end{aligned}$
::3a-9=21

$\begin{aligned} 5 b - 8 = 37 \end{aligned}$
::5b-8=37

$\begin{aligned} 7 x - 10 = 39 \end{aligned}$
::7-10=39

$\begin{aligned} 6 x - 12 = 30 \end{aligned}$
::6-12=30

Write an equation for each word problem and then solve for the unknown variable.
::为每个单词问题写一个方程式, 然后解决未知变量。

Augusta sells t-shirts at the school store. On Tuesday, Augusta sold 7 less than twice the number of t-shirts she sold on Monday. She sold 3 t-shirts on Tuesday. Write an algebraic equation to represent $\begin{aligned} m \end{aligned}$ , the number of t-shirts August sold on Monday.
::Augusta在学校商店出售T恤衫,星期二,Augusta出售7件比星期一少一倍的T恤衫,星期二出售3件T恤衫,写代数方程式代表周一出售的8月T恤。

There are 19 green marbles in a box. The number of green marbles in the box is 6 more than half the number of red marbles in the box. Write an algebraic equation to represent $\begin{aligned} r \end{aligned}$ , the number of red marbles in the box.
::盒子里有19个绿色大理石, 盒子里绿色大理石的数量是盒子里红色大理石数量的一半。写一个代数方程式代表 r, 盒子里红色大理石的数量。

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。

Resources
::资源

Section outline

Solving Two-Step Equations ::解决双层平方

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

Review (Answers) ::回顾(答复)

Resources ::资源

Solving Two-Step Equations
::解决双层平方

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

Review (Answers)
::回顾(答复)

Resources
::资源