Section: 具有类似条件的多级 | 3.5 具有类似用语的多级方

Section outline

Multi-Step Equations with Like Terms
::具有类似条件的多级

When we look at a linear equation we see two kinds of terms : those that contain the unknown variable , and those that don’t. When we look at an equation that has an $\begin{aligned} x \end{aligned}$ on both sides, we know that in order to solve it, we need to get all the $\begin{aligned} x - \end{aligned}$ terms on one side of the equation. This is called combining like terms . The terms with an $\begin{aligned} x \end{aligned}$ in them are like terms because they contain the same variable (or, as you will see in later chapters, the same combination of variables).
::当我们看一个线性方程式时,我们看到两种术语:包含未知变量的术语和不包含变量的术语。当我们看一个在两侧都有 x 的方程式时,我们知道,为了解决这个问题,我们需要将所有 x - 术语都放在方程式的一边。这被称为类似术语的组合。术语与 x 的词是相似的术语,因为它们包含相同的变量(或者,正如您在后面各章所看到的,变量的组合相同 ) 。

Like Terms	Unlike Terms
$\begin{aligned} 4 x, 10 x, - 3.5 x, \end{aligned}$ and $\begin{aligned} \frac{x}{12} \end{aligned}$	$\begin{aligned} 3 x \end{aligned}$ and $\begin{aligned} 3 y \end{aligned}$
$\begin{aligned} 3 y, 0.000001 y, \end{aligned}$ and $\begin{aligned} y \end{aligned}$	$\begin{aligned} 4 x y \end{aligned}$ and $\begin{aligned} 4 x \end{aligned}$
$\begin{aligned} x y, 6 x y, \end{aligned}$ and $\begin{aligned} 2.39 x y \end{aligned}$	$\begin{aligned} 0.5 x \end{aligned}$ and $\begin{aligned} 0.5 \end{aligned}$

Using the Distributive Property of Multiplication
::使用乘法分配属性

To add or subtract like terms, we can use the of Multiplication .
::要增减类似术语,我们可以使用乘法。

\begin{aligned} 3 x + 4 x & = (3 + 4) x = 7 x \\ 0.03 x y - 0.01 x y & = (0.03 - 0.01) x y = 0.02 x y \\ - y + 16 y + 5 y & = (- 1 + 16 + 5) y = 10 y \\ 5 z + 2 z - 7 z & = (5 + 2 - 7) z = 0 z = 0 \end{aligned}

::3x+4x=(3+4)x=7x0.03xy-0.01xy=(0.03-0.01)xy=0.02xy-y+16y+5y=(-1+16+5)y=10y5z+2z-7z=(5+2-7)z=0z=0

To solve an equation with two or more like terms, we need to combine the terms first.
::要用两个或两个以上类似术语来解决等式,我们首先需要将术语结合起来。

Solving for Unknown Values
::解决未知值

1. Solve $\begin{aligned} (x + 5) - (2 x - 3) = 6 \end{aligned}$ .
::1. 解决(x+5)-(2x-3)=6。

There are two like terms: the $\begin{aligned} x \end{aligned}$ and the $\begin{aligned} - 2 x \end{aligned}$ (don’t forget that the negative sign applies to everything in the parentheses). So we need to get those terms together. The associative and distributive properties let us rewrite the equation as $\begin{aligned} x + 5 - 2 x + 3 = 6 \end{aligned}$ , and then the commutative property lets us switch around the terms to get $\begin{aligned} x - 2 x + 5 + 3 = 6 \end{aligned}$ , or $\begin{aligned} (x - 2 x) + (5 + 3) = 6 \end{aligned}$ .
::有两个相似的术语: x 和 ~ 2x (不要忘记负符号适用于括号中的所有字符)。所以我们需要把这些术语合并起来。关联属性和分配属性让我们将方程式重写为 x+5 - 2x+3=6, 然后通量属性让我们转换条件以获得 x-2x+5+3=6 或 (x-2x)+5+3=6 。

$\begin{aligned} (x - 2 x) \end{aligned}$ is the same as $\begin{aligned} (1 - 2) x \end{aligned}$ , or $\begin{aligned} - x \end{aligned}$ , so our equation becomes $\begin{aligned} - x + 8 = 6 \end{aligned}$
:x-2x) 与 (1-2x) 相同, 或 -x, 所以我们的方程式变成 - x+8=6

Subtracting 8 from both sides gives us $\begin{aligned} - x = - 2 \end{aligned}$ .
::从两边减去8后,我们得出了-x2。

And finally, multiplying both sides by -1 gives us $\begin{aligned} x = 2 \end{aligned}$ .
::最后,两边乘以 -1 给我们x=2。

2. Solve $\begin{aligned} \frac{x}{2} - \frac{x}{3} = 6 \end{aligned}$ .
::2. 解决 x2-x3=6。

This problem requires us to deal with fractions. We need to write all the terms on the left over a common denominator of six.
::这个问题要求我们处理分数问题,我们需要在六分之六的共同标准之上写下左侧的所有术语。

\begin{aligned} \frac{3 x}{6} - \frac{2 x}{6} = 6 \end{aligned}

::3x6-2x6=6

Then we subtract the fractions to get $\begin{aligned} \frac{x}{6} = 6 \end{aligned}$ .
::然后我们减去分数以获得 x6=6 。

Finally we multiply both sides by 6 to get $\begin{aligned} x = 36 \end{aligned}$ .
::最后,我们把两边乘以6 以获得x=36。

Example
::示例示例示例示例

Example 1
::例1

Solve $\begin{aligned} \frac{2 x}{5} - \frac{3 x}{2} = 11 \end{aligned}$ .
::解决 2x5 - 3x2=11 。

This problem requires us to deal with fractions. We need to write all the terms on the left over a common denominator of ten.
::这个问题要求我们处理分数问题,我们需要将左侧的所有条款写在10个共同标准之上。

\begin{aligned} \frac{4 x}{10} - \frac{15 x}{10} = 11 \end{aligned}

::4x10-15x10=11

Then we subtract the fractions to get $\begin{aligned} - \frac{11 x}{10} = 11 \end{aligned}$ .
::然后我们减去分数以获得 - 11x10=11。

Finally we multiply both sides by $\begin{aligned} - \frac{10}{11} : \end{aligned}$
::最后,我们把两边乘以-1011:

$\begin{aligned} - \frac{11 x}{10} \cdot - \frac{10}{11} = 11 \cdot - \frac{10}{11} \end{aligned}$
::- 11x101011=111011

to get $\begin{aligned} x = - 10 \end{aligned}$ .
::以获得 x10。

Review
::回顾

Solve the following equations for the unknown variable.
::为未知变量解决以下方程式。

$\begin{aligned} 1.3 x - 0.7 x = 12 \end{aligned}$
::1.3x-0.7x=12

$\begin{aligned} - 10 a - 2 (a + 5) = 14 \end{aligned}$
::-10a-2(a+5)=14

$\begin{aligned} 5 (2 y - 3 y) = - 20 \end{aligned}$
::5( 2y- 3y) @% 20

$\begin{aligned} \frac{2}{3} x - \frac{1}{5} x = \frac{14}{15} \end{aligned}$
::23-15x=1415

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

$\begin{aligned} s - \frac{3 s}{8} = \frac{5}{6} \end{aligned}$
::3s8=56

$\begin{aligned} 10 (y + 5 y) = 10 \end{aligned}$
::10(y+5y)=10

$\begin{aligned} 2.3 x + 2 (0.75 x - 3.5) = 7.5 \end{aligned}$
::2.3x+2(0.75x-3.5)=7.5

$\begin{aligned} 3 (x + 2) + 5 (2 - x) = - 32 \end{aligned}$
::3(x+2)+5(2-x)_____________________________________________________________________________________________________________________________________________________________

$\begin{aligned} 6 x + 2 (5 x - 2) = 12 \end{aligned}$
::6x+2(5x-2)=12

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。

Section outline

Multi-Step Equations with Like Terms ::具有类似条件的多级

Using the Distributive Property of Multiplication ::使用乘法分配属性

Solving for Unknown Values ::解决未知值

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)