节: 检查对等式的解决方案 | 1.8 衡平的检查解决方案

章节大纲

Checking Solutions: Equations
::检查解决方案:等式

You will often need to check solutions to equations in order to check your work. In a math class, verifying that you arrived at the correct solution is very good practice. We check a solution to an equation by replacing the variable in the equation with the value of the solution. A solution should result in a true statement when simplified.
::您通常需要检查方程式的解决方案, 以便检查您的工作。在数学课中, 校验您是否找到了正确的解决方案是非常好的做法。我们检查方程式的解决方案, 将方程式中的变量替换为解决方案的价值。解决方案在简化时应产生真实的语句。

Checking Solutions to Equations
::检查对等式的解决方案

1. Check that the given number is a solution to the equation: $\begin{aligned} y = - 1; 3 y + 5 = - 2 y \end{aligned}$
::1. 检查给定数字是方程的解决方案:y1; 3y+52y

Replace the variable $\begin{aligned} y \end{aligned}$ in the equation with the given value, -1.
::将方程式中的变量 Y 替换为给定值 - 1。

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

::3+5+523(-1)+52(-2-1)-3+5=22=2

2 = 2 is a true statement . This means that $\begin{aligned} y = - 1 \end{aligned}$ is a solution to $\begin{aligned} 3 y + 5 = - 2 y \end{aligned}$ .
::2 = 2 是一个真实的语句。这意味着 y = 1 是 3 + 5 = 2 的解决方案。

2. Check that the given number is a solution to the equation: $\begin{aligned} z = 3; z^{2} + 2 z = 8 \end{aligned}$
::2. 检查给定数字是方程的解决方案:z=3;z2+2z=8

Replace the variable $\begin{aligned} z \end{aligned}$ in the equation with the given value, 3.
::将方程中的变量z 替换为给定值3。

$\begin{aligned} (z)^{2} + 2 (z) & = 8 \\ (3)^{2} + 2 (3) & = 8 \\ 9 + 6 & = 8 \\ 15 & = 8 \end{aligned}$

:z)2+2(z)=8(3)2+2(3)=89+6=815=8

15 = 8 is not a true statement . This means that $\begin{aligned} z = 3 \end{aligned}$ is not a solution to $\begin{aligned} z^{2} + 2 z = 8 \end{aligned}$ .
::15 = 8 不是一个真实的陈述。这意味着 z= 3 不是 z2+2z=8 的解决方案。

Let’s use what we have learned about defining variables, writing equations and writing inequalities to solve some real-world problems.
::让我们利用我们在界定变量、写方程式和写作不平等方面学到的知识来解决一些现实世界的问题。

Real-World Applications
::现实世界应用

Tomatoes cost $0.50 each and avocados cost $2.00 each. Anne buys six more tomatoes than avocados. Her total bill is $8. How many tomatoes and how many avocados did Anne buy? Check your answer!
::Anne买的西红柿比鳄梨多6个。她的总账单是8美元。Anne买的西红柿和鳄梨都有多少?

Define
::定义定义

Let $\begin{aligned} a \end{aligned}$ equal the number of avocados Anne buys.
::让数量相等的鳄梨安妮买。

Translate
::翻译

Anne buys six more tomatoes than avocados. This means that $\begin{aligned} a + 6 \end{aligned}$ is equal to the number of tomatoes.
::Anne比鳄梨多买6个西红柿,这意味着+6等于西红柿的数量。

Tomatoes cost $0.50 each and avocados cost $2.00 each. Her total bill is $8. This means that .50 times the number of tomatoes plus 2 times the number of avocados equals 8.
::番茄每块价格为0.50美元,鳄梨每块成本为2.00美元,其总帐价为8美元,这意味着西红柿数量为50倍,加2倍于鳄梨数量等于8美元。

$\begin{aligned} 0.5 (a + 6) + 2 a & = 8 \\ 0.5 a + 0.5 \cdot 6 + 2 a & = 8 \\ 2.5 a + 3 & = 8 \\ 2.5 a & = 5 \\ a & = 2 \end{aligned}$

::0.5(a+6)+2a=80.5a+0.56+2a=82.5a+3=82.5a=5a=2

Remember that $\begin{aligned} a \end{aligned}$ represents the number of avocados. Since $\begin{aligned} a = 2 \end{aligned}$ , Anne buys two avocados. The number of tomatoes is $\begin{aligned} (a) + 6 = (2) + 6 = 8 \end{aligned}$ .
::记住 a 表示鳄梨的数量。自 a= 2 以来, Anne 购买了 2 鳄梨。番茄的数量是 (a)+6=(2)+6=8。

Answer
::答复答复答复

Anne bought 2 avocados and 8 tomatoes.
::安妮买了两只鳄梨和八只西红柿

Check
::支票支票支票支票

If Anne bought two avocados and eight tomatoes, the total cost would be: $\begin{aligned} (2 \times 2) + (8 \times 0.5) = 4 + 4 = 8 \end{aligned}$ . The answer checks out.
::如果Anne买了两只鳄梨和八只西红柿,总成本为2x2)+(8x0.5)=4+4=8。

Example
::示例示例示例示例

Example 1
::例1

Check that the given number is a solution to the equation: $\begin{aligned} x = - \frac{1}{2}; 3 x + 1 = x \end{aligned}$
::检查给定数字是公式的解决方案 : x\\\ 12; 3x+1=x

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

::3(x)+1=x3(- 12)+1=12(-32)+1*12(-32)+1*12-12*12

$\begin{aligned} - \frac{1}{2} = - \frac{1}{2} \end{aligned}$ is a true statement . This means that $\begin{aligned} x = - \frac{1}{2} \end{aligned}$ is a solution to $\begin{aligned} 3 x + 1 = x \end{aligned}$ .
::- 1212 是真实的语句。这意味着 x12 是 3x+1=x 的解决方案。

Review
::回顾

For 1-9, check whether the given number is a solution to the corresponding equation.
::对于1-9, 请检查给定数字是否是对应方程的解决方案。

$\begin{aligned} a = - 3; 4 a + 3 = - 9 \end{aligned}$
::a3; 4a+3+9

$\begin{aligned} x = \frac{4}{3}; \frac{3}{4} x + \frac{1}{2} = \frac{3}{2} \end{aligned}$
::x=43; 34x+12=32

$\begin{aligned} y = 2; 2.5 y - 10.0 = - 5.0 \end{aligned}$
::y=2; 2.5y-10.05.0

$\begin{aligned} z = - 5; 2 (5 - 2 z) = 20 - 2 (z - 1) \end{aligned}$
::zz5;2(5-2z)=20-2(z-1)

$\begin{aligned} a = 10; 5 a - 7 = 43 \end{aligned}$
::a=10;5a-7=43

$\begin{aligned} x = \frac{2}{3}; 3 x + 5 = 7 \end{aligned}$
::x=23;3x+5=7

$\begin{aligned} x = - 9; \frac{x}{3} \cdot 10 = - 30 \end{aligned}$
::x9; x31030

$\begin{aligned} z = \frac{1}{2}; 2 z = 1.5 - z \end{aligned}$
::z=12; 2z=1.5-z

$\begin{aligned} z = 0.5; z (1 - 2 z) = 6 + 6 (4 z - 3) \end{aligned}$
::z(1-2z)=6+6(4z-3)

The cost of a Ford Focus is 27% of the price of a Lexus GS 450h. If the price of the Ford is $15000, what is the price of the Lexus?
::Ford Focus的成本是Lexus GS 450h价格的27%。如果Ford的价格是1500,那么Lexus的价格是多少?

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

Checking Solutions: Equations ::检查解决方案:等式

Checking Solutions to Equations ::检查对等式的解决方案

Real-World Applications ::现实世界应用

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Checking Solutions: Equations
::检查解决方案:等式

Checking Solutions to Equations
::检查对等式的解决方案

Real-World Applications
::现实世界应用

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)