节: 无乘数解解答系统 | 3.6 无乘数的溶解系统

章节大纲

Three times a number plus 5 equals twice another number. $\begin{aligned} - 4 \end{aligned}$ times the same number minus 2 equals $\begin{aligned} - 2 \end{aligned}$ times the other number. What are the two numbers?
::一个数字乘以3乘以5等于另一个数字的两倍。- 相同数字乘以4乘以减去2等于-2乘以另一个数字。这两个数字是什么?

Solving Systems Without Multiplying
::无乘数解解答系统

In this lesson we will be looking at systems in which the two equations contain coefficients of one variable that are additive inverses (opposites) of one another.
::在这个教训中,我们将研究两个方程式包含一个变量的系数的系统,其中一个变量是彼此的复数添加系数(对数)。

Let's solve the following systems using linear combinations.
::让我们用线性组合解决以下系统。

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

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

Notice that the coefficients of the $\begin{aligned} y \end{aligned}$ terms are opposites. When we add the two equations together, these terms will be eliminated because their sum is $\begin{aligned} 0 y = 0 \end{aligned}$ .
::注意 Y 术语的系数是相反的。当我们将两个方程式加在一起时, 这些参数将被删除, 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 your answer:
::记住要检查您的答案 :

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

Let's solve the following systems using linear combinations.
::让我们用线性组合解决以下系统。

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

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

Notice that the coefficients of the $\begin{aligned} x \end{aligned}$ terms are opposites. When we add the two equations together, these terms will be eliminated because their sum is $\begin{aligned} 0 x = 0 \end{aligned}$ .
::请注意 x 术语的系数是相反的。当我们将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 your answer:
::记住要检查您的答案 :

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

Let's solve the following systems using linear combinations.
::让我们用线性组合解决以下系统。

$\begin{aligned} 2 x + y & = 2 \\ - 3 x + y & = - 18 \end{aligned}$

::2x+y=2-3x+y18

In this case the coefficients of the $\begin{aligned} y \end{aligned}$ terms are the same, not opposites. One way to solve this system using linear combination would be to subtract the second equation from the first instead of adding it. Sometimes subtraction results in more errors, however, particularly when negative numbers are involved. Instead of subtracting, multiply the second equation by -1 and then add them together.
::在此情况下, Y 术语的系数是相同的,而不是相反的。使用线性组合解决这个系统的方法之一是从第一个公式中减去第二个公式,而不是添加第二个公式。但是,有时减法会导致更多的错误, 特别是当涉及负数时。与其减去, 不如将第二个公式乘以-1, 然后将它们加在一起。

$\begin{aligned} - 1 (- 3 x + y & = - 18) \\ 3 x - y & = 18 \end{aligned}$

::-1(-3x+y18)3x-y=18

Essentially, we changed all of the signs of the terms in this equation.
::从根本上说,我们改变了这个等式中术语的所有迹象。

Now we can add the two equations together to eliminate $\begin{aligned} y \end{aligned}$ :
::现在,我们可以把两个方程式加在一起来消除y:

$\begin{aligned} 2 x + y = 2 \\ + \underline{3 x - y = 18} \\ 5 x = 20 \end{aligned}$

::2x+y=2+3x-y=18_5x=20

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

$\begin{aligned} 5 x & = 20 \\ x & = 4 \end{aligned}$

::5x=20x=4

Now that we have found $\begin{aligned} x \end{aligned}$ , plug this value into either equation to find $\begin{aligned} y \end{aligned}$ :
::现在我们找到了 x, 将这个值插入两个方程式中, 以查找 y:

$\begin{aligned} 2 (4) + y & = 2 - 3 (4) + y = - 18 \\ 8 + y & = 2 o r - 12 + y = - 18 \\ y & = - 6 y = - 6 \end{aligned}$

::2(4)+y=2-3(4)+y188+y=2或 -12+y18y=6y6

The solution is therefore: (4, -6).
::因此,解决办法是4,6) 。

Remember to check your answer:
::记住要检查您的答案 :

$\begin{aligned} 2 (4) + (- 6) & = 8 - 6 = 2 \\ - 3 (4) + (- 6) & = - 12 - 6 = - 18 \end{aligned}$

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the two numbers if 3 times a number plus 5 equals twice another number and -4 times the same number minus 2 equals -2 times the other number.
::早些时候,有人要求你找到这两个数字,如果一个数字乘以3乘以5等于另一个数字的两倍,如果-4乘以同一个数字乘以2等于-2乘以另一个数字的两倍。

The system of linear equations represented by this situation is:
::以这种情况为代表的线性方程式系统是:

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

::3x+5=2y-4x-22y

If we add these two equations we get:
::如果我们加上这两个方程式,我们就会得到:

$\begin{aligned} - x + 3 = 0 \end{aligned}$ or $\begin{aligned} x = 3 \end{aligned}$
::- x+3=0或 x=3

We can now substitute $\begin{aligned} x = 3 \end{aligned}$ into either of the original equations to get $\begin{aligned} y = 7 \end{aligned}$ .
::我们现在可以将 x=3 替换为 y= 7 的原始方程式。

Example 2
::例2

Solve the following systems using Linear Combinations.
::使用线性组合解决下列系统。

$\begin{aligned} 4 x + 5 y & = 8 \\ - 2 x - 5 y & = 6 \end{aligned}$

::4x+5y=8-2x-5y=6

First we can add the two equations together to eliminate $\begin{aligned} y \end{aligned}$ and solve for $\begin{aligned} x \end{aligned}$ :
::首先,我们可以将两个方程式加在一起,消除y,解决 x:

$\begin{aligned} 4 x + 5 y = 8 \\ + \underline{- 2 x - 5 y = 6} \\ 2 x = 14 \\ x = 7 \end{aligned}$

::4x+5y=8 2x-5y=6_ 2x=14 x=7

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

$\begin{aligned} 4 (7) + 5 y & = 8 \\ 28 + 5 y & = 8 \\ 5 y & = - 20 \\ y & = - 4 \end{aligned}$

::4(7)+5y=828+5y=85y20y4

Solution: (7, -4)
::解决办法: (7, 4)

Example 3
::例3

Solve the following systems using Linear Combinations.
::使用线性组合解决下列系统。

$\begin{aligned} 2 x + 3 y & = 3 \\ 2 x - y & = 23 \end{aligned}$

::2x+3y=32x-y=23

This time we need to begin by multiplying the second equation by -1 to get $\begin{aligned} - 2 x + y = - 23 \end{aligned}$ . Now we can add the two equations together to eliminate $\begin{aligned} x \end{aligned}$ and solve for $\begin{aligned} y \end{aligned}$ :
::这一次我们首先需要将第二个方程乘以 - 1 以获得 2- 2x+y 23。现在我们可以将两个方程加在一起来消除 x 和解决 y :

$\begin{aligned} 2 x + 3 y = 3 \\ + \underline{- 2 x + y = - 23} \\ 4 y = - 20 \\ y = - 5 \end{aligned}$

::2x+3y=32x+y}23_4y}2x+3y=32x+y}5

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:

$\begin{aligned} 2 x + (3 - 5) & = 3 \\ 2 x - 15 & = 3 \\ 2 x & = 18 \\ x & = 9 \end{aligned}$

::2x+(3-5)=32x-15=32x=18x=9

Solution: (9, -5)
::解决办法9,5)

Example 4
::例4

Solve the following systems using Linear Combinations.
::使用线性组合解决下列系统。

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

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

In this example, the second equation is not written in standard form . We must first rewrite this equation in standard 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 $\begin{aligned} y \end{aligned}$ :
::在此示例中, 第二个方程式不是以标准格式写入的。我们必须首先以标准格式重写此方程式, 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 2- 2x+y2, 在从两边减去 2x 之后。现在我们可以将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)
::解决办法: (0, 2)

Review
::回顾

Solve the following systems using linear combinations.
::使用线性组合解决以下系统。

.

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

.

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

.

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

.

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

.

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

.

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

.

$\begin{aligned} 5 x + 7 y & = 2 \\ 5 x + 3 y & = 38 \end{aligned}$

.

$\begin{aligned} 12 x - 2 y & = 2 \\ 5 x - 2 y & = - 5 \end{aligned}$

.

$\begin{aligned} 2 x + y & = 25 \\ x + y & = 5 \end{aligned}$

.

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

.

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

.

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

.

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

.

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

.

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

Set up and solve a linear system of equations to solve the following word problems.
::设置并解决直线方程式系统, 以解决以下字词问题。

The sum of two numbers is 15 and their difference is 3. Find the two numbers.
::两个数字的总和是15,它们的差异是3 找出这两个数字。

Jessica and Maria got 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 .
::杰西卡和玛丽亚去超市买水果。杰西卡购买了5个苹果和6个橙子,在税前的总额是3.05美元。玛利亚购买了7个苹果和6个橙子,在税前的总额是3.55美元。每种水果的价格是多少? 提示:让x是1个苹果的价格,Y是1个橙子的价格。

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。

章节大纲

Solving Systems Without Multiplying ::无乘数解解答系统

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Solving Systems Without Multiplying
::无乘数解解答系统

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)