Section: 以加号方式解决等号的解决系统 | 4.5 增加的等号溶解系统

Section outline

A football stadium sells regular and box seating. There are twelve times as many regular seats as there are box seats. The total capacity of the stadium is 10,413. How many box seats are in the stadium? How many regular seats?
::一个足球场出售普通座位和箱式座位,正常座位是箱式座位的12倍,体育场的总容量是10,413个,体育场有多少个箱式座位?

We can determine this by using a system of equations . In this section, we discuss another approach to solving systems algebraically.

::我们可以通过一个方程式系统来决定这一点。在本节中,我们讨论另一种用代数法解决系统的方法。

Solving Systems of Equations by Elimination by Addition
::通过加法消除等式的解决系统

In this section, we look at systems in which the two equations contain coefficients of one variable that are additive inverses (opposites) of one another. Recall, the additive inverse property of addition is: $\begin{aligned} a + (- a) = 0 \end{aligned}$ . $\begin{aligned} a \end{aligned}$ and $\begin{aligned} - a \end{aligned}$ are additive inverses.
::在本节中,我们查看两个方程式包含一个变量的系数的系统,其中一个变量是相互的复数(对数),回顾,添加的反数属性是:a+(a)=0.a和-a是复数。

We are going to use additive inverses and equations to take two equations in the system and create one equation with one variable. We can combine the equations, for example,
::我们将使用添加反函数和方程在系统中取两个方程式,然后用一个变量创建一个方程式。例如,我们可以将方程式结合起来,例如,

$\begin{aligned} a = 7 \\ \underline{+ b = 4} \\ a + b = 11 \end{aligned}$
Let's put these two ideas together here.
::a=7+b=4_a+b=11 让我们把这两个想法放在一起。

Example 1
::例1

Solve the system of equations below by elimination by addition:
::解决以下方程式系统,除去以下方程式系统:

$\begin{aligned} 2 x - 3 y & = - 9 \\ 5 x + 3 y & = 30 \end{aligned}$

::2 - 3y% 95x+3y=30

Solution: Notice that the coefficients of the y- terms are additive inverses. When we add the two equations together, these terms will be eliminated because their sum is $\begin{aligned} 0 y = 0 \end{aligned}$ .
::解决方案 : 注意 Y 条件的系数是添加反函数。当我们将这两个方程式加在一起时, 这些条件将被删除, 因为它们的总和是 0y=0 。

$\begin{aligned} 2 x - 3 y = - 9 \\ + \underline{5 x + 3 y = 30} \\ 7 x = 21 \end{aligned}$

::2x-3y9+5x+3y=30_7x=21

Now we can solve for $\begin{aligned} x \end{aligned}$ :
::现在我们可以解决 x:

$\begin{aligned} 7 x & = 21 \\ x & = 3 \end{aligned}$

::7x=21x=3

Now that we have found $\begin{aligned} x \end{aligned}$ , we can plug this value into either equation to find $\begin{aligned} y \end{aligned}$ :
::既然我们找到了 x, 我们可以将这个值插入到 y 的方程式中:

$\begin{aligned} 2 (3) - 3 y & = - 9 5 (3) + 3 y = 30 \\ 6 - 3 y & = - 9 15 + 3 y = 30 \\ - 3 y & = - 15 o r 3 y = 15 \\ y & = 5 y = 5 \end{aligned}$

::2(3)-3y95(3)+3y=306-3y915+3y=30-3y15或3y=15y=5y=5y=5

The solution is therefore : (3, 5).
::因此,解决办法是3,5)

Remember to check the solution :
::记住要检查解决方案 :

$\begin{aligned} 2 (3) - 3 (5) & = 6 - 15 = - 9 \\ 5 (3) + 3 (5) & = 15 + 15 = 30 \end{aligned}$

by CK-12 demonstrates how to use addition to eliminate a variable given stacked equations.
::通过 CK-12 演示如何使用添加来消除给定的堆叠方程式变量。

Example 2
::例2

Solve the system of equations below by elimination by addition.
::解决下面的方程式系统,除去其他方程式。

$\begin{aligned} x + 4 y & = 2 \\ - x - 5 y & = - 3 \end{aligned}$

::x+4y=2 -x-5y3

Solution: Notice that the coefficients of the x- terms are additive inverses. When we add the two equations together, these terms will be eliminated because their sum is $\begin{aligned} 0 x = 0 \end{aligned}$ .
::解答: 注意, x 条件的系数是添加反函数。当我们将这两个方程式加在一起时, 这些条件将被删除, 因为它们的总和是 0x=0 。

$\begin{aligned} x + 4 y = 2 \\ + \underline{- x - 5 y = - 3} \\ - y = - 1 \end{aligned}$

::x+4y=2x-5y3y1

Now we can solve for $\begin{aligned} y \end{aligned}$ :
::现在我们可以解决y:

$\begin{aligned} - y & = - 1 \\ y & = 1 \end{aligned}$

::-y1y=1

Now that we have found $\begin{aligned} y \end{aligned}$ , we can plug this value into either equation to find $\begin{aligned} x \end{aligned}$ :
::既然我们找到了y, 我们可以将这个值插入到两个方程式中, 以找到 x:

$\begin{aligned} x + 4 (1) = 2 - x - 5 (1) = - 3 \\ x + 4 = 2 - x - 5 = - 3 \\ x = - 2 o r - x = 2 \\ x = - 2 \end{aligned}$

::x+4(1)=2 -x-5(1)3 x+4=2 -x-5*3 x%2 或 -x=2 x%2

The solution is therefore: (-2, 1).
::因此,解决办法是-2,1)

Remember to check the solution:
::记住要检查解决方案 :

$\begin{aligned} - 2 + 4 (1) & = - 2 + 4 = 2 \\ - (- 2) - 5 (1) & = 2 - 5 = - 3 \end{aligned}$

Example 3
::例3

Solve the system of equations below by elimination by addition.
::解决下面的方程式系统,除去其他方程式。

$\begin{aligned} 2 x + 3 y & = - 6 \\ y & = 2 x - 2 \end{aligned}$

::2x+3y=6y=2x-2

Solution: The second equation is not written in general form. We must first rewrite this equation in general form so that the variable will align vertically when we add the equations together. The second equation should be $\begin{aligned} - 2 x + y = - 2 \end{aligned}$ after we subtract $\begin{aligned} 2 x \end{aligned}$ from both sides. Now we can add the two equations together to eliminate $\begin{aligned} x \end{aligned}$ and solve for $\begin{aligned} y \end{aligned}$ :
::解析度 : 第二个方程式不是以一般形式写成的。我们必须首先以一般形式重写这个方程式, 这样当我们将方程式加在一起时, 变量会垂直对齐。第二个方程式应该是 ~ 2x+y2, 然后我们从两边减去 2x 。现在我们可以将这两个方程式加在一起来消除 x 并解决 y :

$\begin{aligned} 2 x + 3 y = - 6 \\ + \underline{- 2 x + y = - 2} \\ 4 y = - 8 \\ y = - 2 \end{aligned}$

::2+3y6+- 2x+y2 _ 4y8 y2

Substitute $\begin{aligned} y \end{aligned}$ into one equation to find $\begin{aligned} x \end{aligned}$ :
::将 y 替换为一个方程式以查找 x:

$\begin{aligned} 2 x + 3 (- 2) & = - 6 \\ 2 x - 6 & = - 6 \\ 2 x & = 0 \\ x & = 0 \end{aligned}$

::2x+3( - 2) 62x- 662x=0x=0

Solution: (0, -2).
::解决方案: (0, - 2) 。

Example 4
::例4

A football stadium sells regular and box seating. There are twelve times as many regular seats as there are box seats. The total capacity of the stadium is 10,413. How many box seats are in the stadium? How many regular seats?
::一个足球场出售普通座位和箱式座位,正常座位是箱式座位的12倍,体育场的总容量是10,413个,体育场有多少个箱式座位?

Solution: Let r be the number of regular seats and b be the number of box seats. We get two equations from the information in the problem:
::解答:让 r 代表正常席位的数目, b 代表箱席位的数目。我们从问题的信息中得出两个方程式 :

There are twelve times as many regular seats as there are box seats: $\begin{aligned} r = 12 b \end{aligned}$ .
::正常席位是箱式席位(r=12b)的十二倍。

The total capacity of the stadium is 10,413: $\begin{aligned} r + b = 10, 413 \end{aligned}$ .
::体育场的总容量为10 413人:R+b=10 413人。

In the first equation, we can subtract $\begin{aligned} 12 b \end{aligned}$ and multiply by -1 to create additive inverses.
::在第一个方程中,我们可以减去12b,乘以 -1 来创建添加反函数。

$\begin{aligned} r = 12 b \\ \underline{- 12 b - 12 b} \\ r - 12 b = 0 \end{aligned}$

::r = 12b-12b - 12b_r-12b=0

Notice that none of the coefficients are currently opposites. To fix this we can multiply both sides of the equation by -1.
::请注意, 目前没有一个系数是正对的。要解决这个问题, 我们可以将方程的两边乘以-1 。

$\begin{aligned} - 1 (r - 12 b = 0) \\ - r + 12 b = 0 \end{aligned}$

::-1(r-12b=0)-r+12b=0

Now, we can stack the like terms and eliminate r . Lastly, we substitute into the first equation to find r .
::现在,我们可以堆叠一样的术语并消除r。最后,我们替代的第一个方程找到r。

$\begin{aligned} - r + 12 b = 0 \\ + \underline{r + b = 10, 413} \\ 13 b = 10, 413 \\ b = 801 \\ r = 12 (801) = 9, 612 \end{aligned}$

::-r+12b=0+r+b=10,413_13b=10,413b=10,413b=801r=12(801)=9,612

There are 801 box seats and 9,612 regular seats.
::有801个箱式席位和9 612个普通席位。

Summary
::摘要

To solve a system of equations by elimination by addition, we need to have one pair of terms be additive inverses. Then we add the equations to eliminate that variable.
::为了通过加去除来解决方程式系统,我们需要有一个对等词作为添加反函数。然后我们添加方程式来消除该变量。

Review
::回顾

Solve the following systems using elimination by addition.
::补充消除以下系统。

$\begin{aligned} \begin{aligned} 4 x + 2 y & = - 6 \\ - 5 x - 2 y & = 4 \end{aligned} \end{aligned}$

.

$\begin{aligned} \begin{aligned} - 3 x + 5 y & = - 34 \\ 3 x - y & = 14 \end{aligned} \end{aligned}$

.

$\begin{aligned} \begin{aligned} x + y & = - 1 \\ x - y & = 21 \end{aligned} \end{aligned}$

.

$\begin{aligned} \begin{aligned} 2 x + 8 y & = - 4 \\ - 2 x + 3 y & = 15 \end{aligned} \end{aligned}$

.

$\begin{aligned} \begin{aligned} 8 x - 12 y & = 24 \\ - 3 x + 12 y & = 21 \end{aligned} \end{aligned}$

.

$\begin{aligned} \begin{aligned} x + 3 y & = - 2 \\ - x - 2 y & = 4 \end{aligned} \end{aligned}$

.

$\begin{aligned} \begin{aligned} \frac{1}{2} x + 3 y = - 3 \\ y = \frac{1}{2} x - 5 \end{aligned} \end{aligned}$

.

$\begin{aligned} 3 x + 5 y = 10 \\ y = - 3 x - 10 \end{aligned}$

.

$\begin{aligned} 6 x + 3 y = - 3 \\ 3 y = - 7 x + 1 \end{aligned}$

.

$\begin{aligned} \begin{aligned} 4 x - 2 y & = 5 \\ - 4 x + 2 y & = 11 \end{aligned} \end{aligned}$

.

$\begin{aligned} \begin{aligned} 9 x + 2 y & = 0 \\ - 9 x - 3 y & = 0 \end{aligned} \end{aligned}$

.

$\begin{aligned} 11 x + 7 y = 12 \\ - 11 x = 7 y - 12 \end{aligned}$

Explore More
::探索更多

1. Three times a number plus 5 equals twice another number. Additionally, $\begin{aligned} - 4 \end{aligned}$ times the same number minus 2 equals $\begin{aligned} - 2 \end{aligned}$ times the other number. What are the two numbers?
::1. 一个数字乘以3乘以5等于另一个数字乘以2;此外,一个数字乘以4等于减去2等于-另一个数字乘以2。这两个数字是多少?

2. Jessica and Maria go to the supermarket to buy fruit. Jessica buys 5 apples and 6 oranges and her total before tax is $3.05. Maria buys 7 apples and 6 oranges and her total before tax is $3.55. What is the price of each fruit? Hint: Let $\begin{aligned} x \end{aligned}$ be the price of one apple and $\begin{aligned} y \end{aligned}$ be the price of one orange .
::2. Jessica和Maria去超市买水果,Jessica购买5个苹果和6个橙子,税前总额3.05美元,Maria购买7个苹果和6个橙子,税前总额3.55美元,每块水果的价格是多少?提示:让X是1个苹果的价格,Y是1个橙子的价格。

by CK-12 demonstrates how to solve a system of linear equations by elimination.
::CK-12 显示如何通过删除解决线性方程式系统。

3. Two movie rental stores are in competition. Movie House charges an annual membership of $30 and charges $1.25 per movie rental. Flicks for Cheap charges an annual membership of $15 and charges $2.75 per movie rental. After how many movie rentals would Movie House become the better option?
::电影院每年收取30美元的会员费,每部电影租金收取1.25美元; 廉价的Flicks公司每年收取15美元的会员费,每部电影租金收取2.75美元。

4. A small plane flies from Los Angeles to Denver with a tail wind (the wind blows in the same direction as the plane) and an air-traffic controller reads its ground-speed (speed measured relative to the ground) at 275 miles per hour. Another, identical plane, moving in the opposite direction has a ground-speed of 227 miles per hour. Assuming both planes are flying with identical air-speeds, calculate the speed of the wind.
::4. 一小架飞机从洛杉矶飞到丹佛,尾风(风向与飞机的方向相同),空中交通控制器的地面速度(相对于地面的速度)为每小时275英里,另一架飞机方向相反,地面速度为每小时227英里,假设这两架飞机的飞行速度与飞行速度相同,则计算风速。

5. Paul has a part time job selling computers at a local electronics store. He earns a fixed hourly wage, but can earn a bonus by selling warranties for the computers he sells. He works 20 hours per week. In his first week, he sold eight warranties and earned $260. In his second week, he managed to sell 13 warranties and earned $320. What is Paul’s hourly rate, and how much extra does he get for selling each warranty?
::5. 保罗在当地一家电子商店出售计算机,从事兼职工作,他挣固定的小时工资,但可以通过出售销售的计算机的保修金赚取奖金,每周工作20小时,第一周出售8份保修金,赚取260美元,第二周出售13份保修金,赚取320美元。保罗的小时工资是多少,出售每份保修单能得到多少额外收入?

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try these interactives that reinforce the concepts explored in this section:
::尝试这些强化本节所探讨概念的交互作用 :

Section outline

Solving Systems of Equations by Elimination by Addition ::通过加法消除等式的解决系统

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 for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

Solving Systems of Equations by Elimination by Addition
::通过加法消除等式的解决系统

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 for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX