Section: 单变量 | 7.7 单一变数加上等数

Section outline

The language club is fundraising for a trip to study art and architecture in Paris next summer. They have raised $3,500 from helping the community and $4,800 from various donors. They need a total of $12,300 to subsidize the educational trip. Can you write an equation to solve for how much more they need to raise, and then find out that amount?
::语言俱乐部正在为明年夏天在巴黎的艺术和建筑学习之旅筹集资金。他们从帮助社区募集了3,500美元,从各捐助方募集了4,800美元。他们总共需要12,300美元来补贴教育之旅。你能写一个方程式来解决他们需要筹集多少,然后找出这笔金额吗?

In this concept, you will learn to solve single variable addition equations.
::在此概念中,您将学会解析单变量添加方程式。

Solving Single Variable Addition Equations
::解决单变量增加等量

A variable is used to represent a number, quantity, or expression .
::变量用于表示数字、数量或表达式。

For example, in the algebraic equation below, the variable $\begin{aligned} x \end{aligned}$ represents one possible number.
::例如,在下面的代数方程式中,变量x代表一个可能的数字。

$\begin{aligned} x + 3 = 5 \end{aligned}$

::x+3=5 x+3=5

To find out what number $\begin{aligned} x \end{aligned}$ represents, ask yourself, “What number, when added to 3, equals 5?”
::要知道数字x代表什么,请问您自己,“如果加上3,数字是5,数字是多少?”

$\begin{aligned} 2 + 3 = 5 \end{aligned}$ , so $\begin{aligned} x \end{aligned}$ must be equal to 2.
::2+3=5, 所以x必须等于 2 。

When solving more complex equations, such as $\begin{aligned} x + 34 = 72 \end{aligned}$ , it is important to be more systematic and strategic.
::在解决更复杂的方程式,如x+34=72时,必须更加系统和战略性。

To solve an equation you should work to isolate the variable . Isolating the variable means getting the variable by itself on one side of the equal ( $\begin{aligned} = \end{aligned}$ ) sign.
::要解析方程式, 您应该工作孤立变量。孤立变量意味着在等号( =) 的一边获得变量。

One way to isolate the variable is to use an inverse operation , that is, the ‘opposite’ operation . For example, addition is the inverse of subtraction , subtraction is the inverse of addition, multiplication is the inverse of division , and division is the inverse of multiplication.
::孤立变量的一种方法是使用反向操作,即“对面”操作。例如,加法是减法的反反向,减法是反向,乘法是反向的,乘法是反向的,除法是反向的,而除法是反向的乘法。

To solve an equation in which a variable is added to a number, you can use the inverse of addition––subtraction.
::要解析将变量添加到数字中的方程式, 您可以使用反向的增- 减- 减- 减。

To do this you need to use the Subtraction Property of Equality , which states: if $\begin{aligned} a = b \end{aligned}$ , then $\begin{aligned} a - c = b - c \end{aligned}$ .
::要做到这一点,你需要使用平等减法财产,其中规定:如果a=b,a-c=b-c。

This means that if you subtract a number, $\begin{aligned} c \end{aligned}$ , from one side of an equation, you must subtract that same number, $\begin{aligned} c \end{aligned}$ , from the other side, too, to keep the values on both sides equal.
::这意味着,如果从公式的一面减去一个数字,c,则必须从另一面减去同一个数字,c,以保持双方的数值相等。

Let’s look at an example.
::让我们举个例子。

Solve for $\begin{aligned} x \end{aligned}$ .
::解决x。

$\begin{aligned} x + 34 = 72 \end{aligned}$
::x+34=72

Use the subtraction property of equality to subtract 34 from both sides of the equation. This will isolate the variable $\begin{aligned} x \end{aligned}$ .
::使用等值的减法属性从方程两侧减去34。这将分离变量 x。

$\begin{aligned} \begin{array}{rcl} x + 34 & = & 72 \\ x + 34 - 34 & = & 72 - 34 \\ x + 0 & = & 38 \\ x & = & 38 \end{array} \end{aligned}$
The answer is $\begin{aligned} x = 38 \end{aligned}$ .
::x+34=72x+34-34=72-34x+038x=38 回答是 x=38。

Here is another example.
::下面是另一个例子。

Solve for $\begin{aligned} b \end{aligned}$ .
::解决b.

$\begin{aligned} 1.5 + b = 3.5 \end{aligned}$
::1.5+b=3.5

In the equation, 1.5 is added to $\begin{aligned} b \end{aligned}$ . So, use the subtraction property of equality and subtract 1.5 from both sides of the equation to solve for $\begin{aligned} b \end{aligned}$ .
::b. 因此,使用等值减法属性,减去公式两侧的1.5,解决b。

$\begin{aligned} \begin{array}{rcl} 1.5 + b & = & 3.5 \\ 1.5 - 1.5 + b & = & 3.5 - 1.5 \\ 0 + b & = & 2.0 \\ b & = & 2 \end{array} \end{aligned}$
The answer is $\begin{aligned} b = 2 \end{aligned}$ .
::1.5+b=3.51.5-1.5+b=3.5-1.50+b=2.0b=2 答案是b=2。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the language club, who is going to study art and architecture in Paris.
::早些时候,你被问及语言俱乐部的问题, 他将在巴黎学习艺术和建筑。

They have raised $3,500 from one source and $4,800 from another. However, they need a total of $12,300 for the educational trip. Write an equation to solve for how much more they need to raise.
::他们从一个来源筹集了3,500美元,从另一个来源筹集了4,800美元,然而,他们需要总共12,300美元用于教育旅行。写一个方程式以解决他们需要筹集多少资金的问题。

First, let $\begin{aligned} x \end{aligned}$ be how much money the students still need to raise.
::首先,让我们先看看学生们还需要筹集多少钱。

Next, you need to translate the language into a mathematical equation. Add up all of the funding plus what they still need to raise, $\begin{aligned} x \end{aligned}$ , and set that equal to the total amount needed.
::接下来, 您需要将语言转换为数学等式。添加所有资金加上他们仍需要筹集的资金, x, 并设定与所需总额相等。

$\begin{aligned} 3, 500 + 4, 800 + x = 12, 300 \end{aligned}$

::3 500+4 800+x=12 300

Next, add the numbers together.
::下一步,将数字加在一起。

$\begin{aligned} 8, 300 + x = 12, 300 \end{aligned}$

::8 300+x=12 300美元

Now, use the subtraction property of equality to subtract 8,300 from both sides of the equation. This isolates the variable $\begin{aligned} x \end{aligned}$ .
::现在,使用等值的减法属性从方程的两侧减去8,300美元。这将变量 x 分离出来。

$\begin{aligned} \begin{array}{rcl} 8, 300 - 8, 300 + x & = & 12, 300 - 8, 300 \\ x & = & 4, 000 \end{array} \end{aligned}$

::8,300-8,300,300+x=12,300-8,300x=4,000

The answer is the students still need to raise $4,000.
::答案是学生仍需要筹募4000美元。

Example 2
::例2

The number of gray tiles in a bag is 4 more than the number of blue tiles in the bag. There are 11 gray tiles in the bag.
::袋子中的灰砖数比袋子中的蓝砖数多4个,袋子中有11个灰砖。

Write an equation to represent $\begin{aligned} b \end{aligned}$ , the number of blue tiles in the bag, and then find the value of $\begin{aligned} b \end{aligned}$ .
::写一个方程式以表示 b, 包中蓝色瓷砖的数, 然后找到 b 的值。

First, translate the language into a mathematical equation. “Is ” means equals, $\begin{aligned} b \end{aligned}$ is the number of blue tiles, and there are 11 gray tiles.
::首先,将语言翻译成数学等式。“是”的意思是等同的,b是蓝色瓷砖的数量,有11块灰砖。

$\begin{aligned} The \underline{number of gray tiles} \dots \underline{is 4 more than} the \underline{number of blue tiles} \dots \\ ↓ ↓ ↓ ↓ \\ 11 = 4 + b \end{aligned}$

::灰色瓷砖的数_... 是蓝色瓷砖的数_ 4 大于_ 蓝色瓷砖的数_... @ @ @ @ @ @ @ @ @ @ @ @ @11=4+b

So the equation is $\begin{aligned} 11 = 4 + b \end{aligned}$ .
::等式是11=4+b。

Solve the equation to find the number of blue tiles in the bag. Use the subtraction property of equality, and subtract 4 from each side of the equation. This isolates the variable.
::解析方程式以在包中找到蓝色瓷砖的数量。使用等式的减法属性, 然后从方程式的每侧减去 4 。这样可以分离变量。

$\begin{aligned} \begin{array}{rcl} 11 & = & 4 + b \\ 11 - 4 & = & 4 - 4 + b \\ 7 & = & 0 + b \\ 7 & = & b \end{array} \end{aligned}$

::11=4+B11-4=4-4-4-4+B7=0+B7=b

The answer is there are 7 blue tiles in the bag.
::答案是袋子里有七块蓝色瓷砖

Solve each addition equation for the missing variable.
::解决缺失变量的每个添加方程式。

Example 3
::例3

$\begin{aligned} x + 36 = 90 \end{aligned}$
::x+36=90

Use the subtraction property of equality and subtract 36 from both sides of the equation. This isolates the variable $\begin{aligned} x \end{aligned}$ .
::使用等值减法的减法属性,从方程两侧减去36。此选项将变量 x 分离出来。

$\begin{aligned} \begin{array}{rcl} x + 36 & = & 90 \\ x + 36 - 36 & = & 90 - 36 \\ x + 0 & = & 54 \\ x & = & 54 \end{array} \end{aligned}$

::x+36=90x+36=36=90-36=36x+054x=54

The answer is $\begin{aligned} x = 54 \end{aligned}$ .
::答案是 x=54 。

Example 4
::例4

$\begin{aligned} x + 27 = 35 \end{aligned}$
::x+27=35

Use the subtraction property of equality and subtract 27 from both sides of the equation. This isolates the variable $\begin{aligned} x \end{aligned}$ .
::使用等值减法的减法属性,从方程两侧减去27。此选项将变量 x 分离出来。

$\begin{aligned} \begin{array}{rcl} x + 27 & = & 35 \\ x + 27 - 27 & = & 35 - 27 \\ x & = & 8 \end{array} \end{aligned}$

::x+27=35x+27=27=35-27x=8

The answer is $\begin{aligned} x = 8 \end{aligned}$ .
::答案是 x=8 。

Example 5
::例5

$\begin{aligned} y + 1.7 = 6.5 \end{aligned}$
::y+1.7=6.5 y+1.7=6.5

Use the subtraction property of equality and subtract 1.7 from both sides of the equation. This isolates the variable $\begin{aligned} y \end{aligned}$ .
::使用等值减法的减法属性,从方程两侧减去1.7。此选项将变量 y 分隔开。

$\begin{aligned} \begin{array}{rcl} y + 1.7 & = & 6.5 \\ y + 1.7 - 1.7 & = & 6.5 - 1.7 \\ y & = & 4.8 \end{array} \end{aligned}$

::y+1.7=6.5y+1.7-1.7=6.5-1.7y=4.8 y+6.5y+1.7-1.7=6.5-1.7y=4.8

The answer is $\begin{aligned} y = 4.8 \end{aligned}$ .
::答案是y=4.8

Review
::回顾

Solve each single-variable addition equation.
::解决每个单变量附加方程式。

$\begin{aligned} x + 7 = 14 \end{aligned}$
::x+7=14 x7=14

$\begin{aligned} y + 17 = 34 \end{aligned}$
::y+17=34

$\begin{aligned} a + 27 = 34 \end{aligned}$
::a+27=34

$\begin{aligned} x + 30 = 47 \end{aligned}$
::x+30=47

$\begin{aligned} x + 45 = 53 \end{aligned}$
::x+45=53

$\begin{aligned} x + 18 = 24 \end{aligned}$
::x+18=24

$\begin{aligned} a + 38 = 74 \end{aligned}$
::a +38=74

$\begin{aligned} b + 45 = 80 \end{aligned}$
::b+45=80

$\begin{aligned} c + 54 = 75 \end{aligned}$
::c+54=75

$\begin{aligned} y + 197 = 423 \end{aligned}$
::y+197=423 y+197=423

$\begin{aligned} y + 297 = 523 \end{aligned}$
::y+297=523

$\begin{aligned} y + 397 = 603 \end{aligned}$
::y+397=603

$\begin{aligned} y + 97 = 405 \end{aligned}$
::y+97=405

$\begin{aligned} y + 94 = 102 \end{aligned}$
::y+94=102 y+94=102

$\begin{aligned} y + 87 = 323 \end{aligned}$
::y+87=323 y+87=323

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 Single Variable Addition Equations ::解决单变量增加等量

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 Single Variable Addition Equations
::解决单变量增加等量

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