Section: 解析多级 | 1.8 解决多层多层

Section outline

Introduction
::导言

In this year’s election for student government president, there were two candidates. The winner received $\begin{align*}\frac{1}{3}\end{align*}$ more votes than the loser. If there were 584 votes cast for president, how many votes did each of the two candidates receive?
::在今年的中学生政府主席选举中,有两名候选人。获胜者比失败者多得13票。如果有584张选票投给总统,那么两位候选人每人获得多少选票?

Solving Equations
::溶解等量

To solve the above problem, you will need to create an equation and solve it using multiple steps.
::要解决上述问题, 您需要创建方程式, 并使用多个步骤来解决。

One-Step Equations
::单步等量

The goal of solving an equation is to get the variable by itself on one side of the equation.
::解决方程式的目标是将变量自己放在方程式的一边

In order to do this, you need to remember two main approaches:
::要做到这一点,你需要记住两个主要方法:

Use properties of equality to keep equations balanced.
::使用平等属性来保持方程式平衡。

Use inverse operations .
::使用反向操作。

Properties of Equality
::平等财产

For real numbers $\begin{align*}a\end{align*}$ , $\begin{align*}b\end{align*}$ , and $\begin{align*}c\end{align*}$ :
::对于a、b和c的实际数字:

Addition Property of Equality
If $\begin{align*}a=b\end{align*}$ , then $\begin{align*}a+c=b+c\end{align*}$ .
::如果a=b,然后a+c=b+c,则a+c=b+c。

Subtraction follows from addition :
If $\begin{align*}a=b\end{align*}$ , then $\begin{align*}a+(\text{-}c) = b+(\text{-}c)\end{align*}$ or $\begin{align*}a-c=b-c\end{align*}$ .
::如果a=b,则a+(-c)=b+(-c)或a-c=b-c。

Multiplication Property of Equality :
If $\begin{align*}a=b\end{align*}$ , then $\begin{align*}a \times c = b \times c\end{align*}$ .
::平等乘数属性:如果a=b,那么axc=bxc。

Division follows from multiplication :
If $\begin{align*}a=b\end{align*}$ , then $\begin{align*}a \times \tfrac{1}{c} = b \times \tfrac{1}{c}\end{align*}$ or $\begin{align*}\tfrac{a}{c}=\tfrac{b}{c}\end{align*}$ for $\begin{align*}c\neq 0\end{align*}$ .
::如果 a=b,那么ax1c=bx1c或ac=bc表示c=0。

Multi-Step Equations
::多级

To solve equations with multiple steps, use the reverse of the to help you solve. For more complex equations, combine like terms . Like terms are terms with the same variables with matching exponents.
::要用多个步骤解析方程式, 请使用反向来帮助您解析。对于更复杂的方程式, 将类似条件合并。类似条件的术语是用相同的变量和对应的指数来解析的术语。

For example, $\begin{align*}3x^2\end{align*}$ and $\begin{align*}x^2\end{align*}$ are like terms because they both have the same variable and exponent . Similarly, $\begin{align*}xy^3z^2\end{align*}$ and $\begin{align*}-4xy^3z^2\end{align*}$ are also like terms. However, $\begin{align*}xy^2\end{align*}$ and $\begin{align*}x^2y\end{align*}$ are not like terms because the exponents are switched based on the variables.
::例如, 3x2 和 x2 是相似的术语, 因为它们都具有相同的变量和引言。同样, xy3z2 和 - 4xy3z2 也类似术语。但是, xy2 和 x2 与 3x2 和 x2 不同, 因为指数是根据变量转换的。

Equations with Fractions
::带有分数的平方

When introducing fractions into an equation, the same rules for solving any equation apply. You need to keep the equations in balance by adding, subtracting, multiplying, or dividing on both sides of the equal sign in order to isolate the variable . The goal still remains to get your variable alone on one side of the equal sign, with your constant terms on the other, to solve for this variable. If there are multiple fractions that don't have the same denominator, you must 1st find the least common denominator (LCD) before combining like terms. Or, sometimes you need to multiply or divide the equation by the numerator and denominator to solve for the variable.
::当将分数引入方程式时, 应用相同的解决公式的规则。您需要通过在等号的两侧添加、减、乘或分隔来保持方程式的平衡, 以便孤立变量。目标仍然是将变量单放在等号的一边, 并用您的不变条件解决此变量。如果有多个分数没有相同的分母, 您就必须在合并类似条件之前先找到最小的公分母( LCD ) 。或者, 有时您需要将公式乘以数字和分母, 才能解开变量。

Examples
::实例

Example 1
::例1

Solve $\begin{align*}k-5 = 12\end{align*}$ .
::解决 k - 5= 12 。

Solution:
::解决方案 :

Since the inverse of subtraction is addition, add 5 to both sides of the equation:
::由于加上减去的反数,方程两侧加上5:

$\begin{aligned} k - 5 = 12 \\ \underline{+ 5 + 5} \end{aligned}$
$\begin{align*}& k-5=12\\ & \underline{ \ \ +5 \ \ +5}\end{align*}$
::k-5=12+5+5_

The additive inverse property notes that the sum of a number and its opposite equal zero, so $\begin{align*}-5+5 = 0\end{align*}$ , which will cancel out the number next to the variable. The addition property of equality allows you to add a number to a side, as long as you add the same number to both sides:
::添加反向属性指出,一个数字的总和及其对数等于零, 也就是- 5+5=0, 这将取消变量旁边的数字。添加平等属性允许您在侧面添加一个数字, 只要您在两侧都添加相同数字 :

$\begin{aligned} k - 5 = 12 \\ \underline{+ 5 + 5} \\ k = 17 \end{aligned}$
$\begin{align*}& k\cancel{-5}=12\\ & \underline{\ \ \cancel{+5} \ \ +5}\\ & \quad \ \ \ k=17\end{align*}$
::k-5=12+5+5+5_k=17

Example 2
::例2

Solve $\begin{align*}\frac{m}{4}=3\end{align*}$ .
::解决 m4=3 。

Solution:
::解决方案 :

The inverse of division is multiplication, so multiply by 4 on both sides of the equation:
::分裂的反面是乘法, 方程式两侧乘以4:

$\begin{aligned} 4 \times \frac{m}{4} = 3 \times 4 \end{aligned}$
$\begin{align*}4\times \frac{m}{4}=3 \times 4\end{align*}$ The multiplicative inverse property notes that $\begin{align*}4 \times \tfrac{1}{4} = 1\end{align*}$ , which will cancel out the number under the variable. The multiplication property of equality allows you to multiply a side by a number, as long as you multiply both sides by the same number:
::4xm4=3x4 多倍反向属性指出, 4x14=1, 这将取消变量下的数字。等值的倍增属性允许您将一边乘以数, 只要您将两边乘以相同数字 :

$\begin{aligned} 4 \times \frac{m}{4} & = 3 \times 4 \\ m & = 12 \end{aligned}$
$\begin{align*}\cancel{4}\times \frac{m}{\cancel{4}}&=3 \times 4\\ m&=12\end{align*}$
::4xm4=3x4m=12

Example 3
::例3

Solve $\begin{align*}3x + 5 = 11\end{align*}$ .
::解决 3x+5=11 。

Solution:
::解决方案 :

If we were evaluating this, we would multiply and then add. However, when solving the equation for x, start by working backwards by first dealing with the addition and then the multiplication. Subtract 5 from both sides of the equation to cancel out the addition by the additive inverse property:
::如果我们评估了这一点, 我们就会乘数然后添加。但是, 当解答 x 的方程时, 首先从倒向工作开始, 首先是处理添加, 然后是乘数。从方程的两边减去 5 来取消添加的反属性 :

$\begin{aligned} 3 x + 5 = 11 \\ \underline{- 5 - 5} \\ 3 x = 6 \end{aligned}$
$\begin{align*}& 3x+\cancel{5}=11\\ & \underline{\quad \ \ \cancel{\text{-}5} \ \quad \text{-}5 \ }\\ & \qquad 3x = 6\end{align*}$
::3x+5=11-5-5-5_3x=6

Divide 3 from both sides of the equation to cancel out the multiplication by the multiplicative inverse property :
::从等式的两侧除以 3 , 以取消乘法的乘法 :

$\begin{aligned} \underline{3 x} = \underline{6} \\ 3 3 \\ x = 2 \end{aligned}$
$\begin{align*}& \underline{\cancel{3}x} = \underline{6}\\ & \ \ \cancel{3} \quad \ 3\\ & \quad x = 2 \end{align*}$
::3x6_3 3x=2

Example 4
::例4

Solve $\begin{align*}5x-10 = 3x + 14\end{align*}$ .
::解决 5x- 10=3x+14。

Solution:
::解决方案 :

Bring like terms to the same side and combine, using inverse operations:
::使用反向操作,将类似条件带往同一侧并结合:

$\begin{aligned} 5 x - 10 = 3 x + 14 \\ \underline{- 3 x - 3 x} \\ 2 x - 10 = 14 \end{aligned}$
$\begin{align*}& \ \ 5x-10 = \cancel{3x} + 14\\ & \underline{-3x \ \ \qquad \cancel{-3x} \ \ \ \ \quad}\\ & \ \ 2x - 10 = 14\end{align*}$ Now solve as before:
::5- 10= 3x+14-3x- 3x_ 2x- 10= 14 现在和以前一样解析 :

$\begin{aligned} 2 x - 10 = 14 \\ \underline{+ 10 + 10} \\ \underline{2 x} = \underline{24} \\ 2 2 \\ x = 12 \end{aligned}$
$\begin{align*}& 2x-\cancel{10}=14\\ & \underline{\quad +\cancel{10} \ +10}\\ & \quad \ \ \ \underline{\cancel{2}x}=\underline{24}\\ & \qquad \cancel{2} \quad \ \ 2\\ & \qquad \ \ x=12\end{align*}$
::2 - 10=14+10+10+10_ 2x_ 24_ 2x2x2x=12

Example 5
::例5

Solve $\begin{align*}\frac{1}{3}t+5=-1\end{align*}$ .
::解决 13t+5+1 。

Solution:
::解决方案 :

$\begin{aligned} \frac{1}{3} t + 5 & = - 1 \\ \frac{1}{3} t + 5 - 5 & = - 1 - 5 & (Subtract 5 from both sides to isolate the variable .) \\ \frac{1}{3} t & = - 6 & (Simplify .) \\ (3) \frac{1}{3} t & = - 6 (3) & (Multiply both sides by the denominator (3) in the fraction .) \\ t & = - 18 & (Simplify.) \end{aligned}$
$\begin{align*}\frac{1}{3}t+5 &= -1\\ \frac{1}{3}t+5 {\color{red}-5} &= -1 {\color{red}-5} && ( \text{Subtract} \ 5 \ \text{from both sides to isolate the variable}.)\\ \frac{1}{3}t &= -6 && ( \text{Simplify}.)\\ ({\color{red}\cancel{3}}) \frac{1}{\cancel{3}}t &= -6 ({\color{red}3}) && ( \text{Multiply both sides by the denominator} \ ({\color{red}3}) \ \text{in the fraction}.)\\ t &= -18 && ( \text{Simplify.})\end{align*}$
::13t+5113t+5-51-5(从两边抽取5分解变量)13t6(简化)(3)13t6(3)(按分数的分数(3)分母截取双方。)t18(简化)。

Therefore , $\begin{align*}t = -18\end{align*}$ .
::因此,T18。

$\begin{aligned} Check: \\ \frac{1}{3} t + 5 & = - 1 \\ \frac{1}{3} (- 18) + 5 & = - 1 \\ - 6 + 5 & = - 1 \\ - 1 & = - 1 \end{aligned}$
$\begin{align*}\text{Check:} &\\ \frac{1}{3}t+5 &= -1\\ \frac{1}{3} ({\color{red}-18})+5 &= -1\\ -6+5 &= -1\\ -1 &= -1 \ \ \end{align*}$
::检查: 13t+5113( - 18)+5}1 - 6+5}1 - 11

Example 6
::例6

Solve $\begin{align*}\frac{2}{5}w-4=-\frac{1}{5}w+8\end{align*}$ .
::解决 25w -415w+8。

Solution:
::解决方案 :

$\begin{aligned} \frac{2}{5} w - 4 & = - \frac{1}{5} w + 8 \\ \frac{2}{5} w - 4 + 4 & = - \frac{1}{5} w + 8 + 4 \\ \frac{2}{5} w & = - \frac{1}{5} w + 12 \\ \frac{2}{5} w + \frac{1}{5} w & = - \frac{1}{5} w + \frac{1}{5} w + 12 \\ \frac{3}{5} w & = 12 \\ \frac{3}{5} w \cdot \frac{5}{3} & = 12 \cdot \frac{5}{3} \\ w & = 20 \end{aligned}$
$\begin{align*}\frac{2}{5}w-4 &= -\frac{1}{5}w+8\\ \frac{2}{5}w-4 {\color{red} \ + \ 4} &= -\frac{1}{5}w +8{\color{red} \ + \ 4} \\ \frac{2}{5}w &= -\frac{1}{5}w+12 \\ \frac{2}{5}w {\color{red} \ + \ \frac{1}{5}w} &= -\frac{1}{5}w {\color{red} \ + \ \frac{1}{5}w}+12 \\ \frac{3}{5}w &= 12 \\ \frac{3}{5}w {\color{red} \ \cdot \ \frac{5}{3}}&= 12 {\color{red} \ \cdot \ \frac{5}{3}}\\ w &= 20 \end{align*}$
::25+4+4+4+8w+8+4+4+15w+8+4+415w+8+415w+1225w+15w+15w+1235w+15w+1235w=1235w+53=12+53w=20

Therefore, $\begin{align*}w = 20\end{align*}$ .
::因此,W=20。

You can always check your answer by plugging it in and evaluating both sides to make sure they are equal.
::你总是可以检查你的答案通过插插插和对两边的评价以确保它们平等。

$\begin{aligned} \frac{2}{5} w - 4 & = - \frac{1}{5} w + 8 \\ \frac{2}{5} (20) - 4 & = - \frac{1}{5} (20) + 8 \\ \frac{40}{5} - 4 & = - \frac{20}{5} + 8 \\ 8 - 4 & = - 4 + 8 \\ 4 & = 4 \end{aligned}$
$\begin{align*}\frac{2}{5}w-4 &= -\frac{1}{5}w+8\\ \frac{2}{5} ({\color{red}20})-4 &=- \frac{1}{5} ({\color{red}20})+8\\ \frac{40}{5}-4 &=-\frac{20}{5}+8\\ 8-4 &= -4+8\\ 4 &= 4 \ \ \end{align*}$
::25 - 415w+825(20) - 415(20)+8405 - 4205 - 205+88 - 44+84=4

Example 7
::例7

Earlier, you were given a problem about a student election for president:
::先前,你曾遇到学生选举总统的问题:

In this year’s student election for president, there were two candidates. The winner received $\begin{align*}\frac{1}{3}\end{align*}$ more votes. If there were 584 votes cast for president, how many votes did each of the two candidates receive?
::在今年的学生总统选举中,有两个候选人。获胜者又获得了13张选票。如果有584张选票投给总统,那么这两名候选人每人得票多少票?

Solution:
::解决方案 :

Let $\begin{align*}x =\end{align*}$ votes for candidate 1 (the winner)
::让我们x=对候选人1(获胜者)的投票

Let $\begin{align*}y =\end{align*}$ votes for candidate 2
::让我们以y=y=票选举候选人2

$\begin{align*}x + y = 584\end{align*}$
::x+y=584

You must have only one variable in the equation in order to solve it. Let’s look at another relationship from the problem:
::您必须在方程中只有一个变量才能解决它。让我们从问题的角度来审视另一个关系:

$\begin{aligned} x & = y + \frac{1}{3} y & (Candidate 1 received \frac{1}{3} more votes than candidate 2.) \\ x & = \frac{3}{3} y + \frac{1}{3} y & (Make denominator common for both y variables .) \\ x & = \frac{4}{3} y & (Simplify.) \end{aligned}$
$\begin{align*}x &= y + \frac{1}{3} y && (\text{Candidate} \ 1 \ \text{received} \ \frac{1}{3} \ \text{more votes than candidate} \ 2.)\\ x &= \frac{3}{3} y + \frac{1}{3} y && ( \text{Make denominator common for both} \ y \ \text{variables}.)\\ x &= \frac{4}{3} y && ( \text{Simplify.})\end{align*}$
::xy+13y( Capidate 1 获得的选票比候选人2.)x=33y+13y( 两个变量 y.)x=43y( 简单化) 常见的 make clex 。)

Now substitute into the original problem and simplify:
::现改为原问题并简化:

$\begin{aligned} \frac{4}{3} y + y & = 584 \\ \frac{4}{3} y + \frac{3}{3} y & = 584 \\ \frac{7}{3} y & = 584 \\ (3) \frac{7}{3} y & = 584 (3) \\ 7 y & = 1752 \\ \frac{7 y}{7} & = \frac{1752}{7} \\ y & = \frac{1752}{7} \end{aligned}$
$\begin{align*}\frac{4}{3} y + y &= 584 \\ \frac{4}{3} y + \frac{3}{3} y &= 584 \\ \frac{7}{3} y &= 584 \\ {\color{red}(\cancel{3})} \frac{7}{\cancel{3}} y &= 584 {\color{red}(3)} \\ 7y &= 1752 \\ \frac{\cancel{7} y}{{\color{red}\cancel{7}}} &= \frac{1752}{{\color{red}7}} \\ y &= \frac{1752}{7} \end{align*}$
::43+y=58443y+33y=58473y=584(3)73y=584(3)7y=584(3)7y=17527y7=17527y=17527y=17527

So, candidate 2 received 250 votes. Candidate 1 would then receive $\begin{align*}584 - 250 = 334\end{align*}$ votes.
::因此,候选人2获得250票,候选人1获得584-250=334票。

Example 8
::例8

Solve for x: $\begin{align*}\frac{2}{3}x=12\end{align*}$ .
::x: 23x=12, 解决。

Solution:
::解决方案 :

$\begin{aligned} \frac{2}{3} x & = 12 \\ (3) \frac{2}{3} x & = 12 (3) \\ 2 x & = 36 \\ \frac{2 x}{2} & = \frac{36}{2} \\ x & = 18 \end{aligned}$
$\begin{align*}\frac{2}{3}x &= 12\\ ({\color{red}\cancel{3}}) \frac{2}{3}x &= 12 ({\color{red}3}) \\ 2x &= 36 \\ \frac{\cancel{2} x}{{\color{red}\cancel{2}}} &= \frac{36}{{\color{red}2}} \\ x &= 18 \end{align*}$
::23x=12(3)23x=12(3)2x=362x2=362x=18

Therefore, $\begin{align*}x = 18\end{align*}$ .
::因此,x=18。

$\begin{aligned} Check: \\ \frac{2}{3} x & = 12 \\ \frac{2}{3} (18) & = 12 \\ \frac{36}{3} & = 12 \\ 12 & = 12 \end{aligned}$
$\begin{align*}\text{Check:} &\\ \frac{2}{3}x &= 12\\ \frac{2}{3} ({\color{red}18}) &= 12\\ \frac{36}{3} &= 12\\ 12 &= 12 \ \ \end{align*}$
::检查: 23x= 1223( 18) = 12363= 1212= 1212

Review
::回顾

Solve for the variable in each of the following equations:
::解决以下方程式中每个方程式中的变量 :

$\begin{align*}\frac{1}{3}p=5\end{align*}$
::13p=5

$\begin{align*}\frac{3}{7}j=8\end{align*}$
::37j=8

$\begin{align*}2b+4=6\end{align*}$
::2b+4=6

$\begin{align*}\frac{2}{3}x-2=1\end{align*}$
::23x-2=1

$\begin{align*}x+3=-3\end{align*}$
::x+3+%3

$\begin{align*}k+\frac{2}{3}=5k\end{align*}$
::k+23=5k (k+23=5k)

$\begin{align*}\frac{1}{6}c+\frac{1}{3}=-2\end{align*}$
::16c+132

$\begin{align*}4x+3=19\end{align*}$
::4x+3=19

$\begin{align*}\frac{3}{4}x-\frac{2}{5}=\frac{1}{2}\end{align*}$
::34 - 25=12

$\begin{align*}\frac{t}{4}+ 3=2\end{align*}$
::t4+3=2

$\begin{align*}\frac{1}{3}x+\frac{1}{4}x=1\end{align*}$
::13x+14x=1

$\begin{align*}d+2d=\frac{5}{3}\end{align*}$
::d+2d=53

$\begin{align*}7x-1=3x-5\end{align*}$
::7x-1=3x-5

$\begin{align*}\frac{1}{3}x-\frac{1}{2}=\frac{3}{4}x\end{align*}$
::13 - 12=34x

$\begin{align*}\frac{2}{3}j-\frac{1}{2}=\frac{3}{4}j+\frac{1}{3}\end{align*}$
::23-12=34j+13

Review ( Answers)
::回顾(答复)

Please see the Appendix.
::请参看附录。

Section outline

Introduction ::导言

Solving Equations ::溶解等量

One-Step Equations ::单步等量

Properties of Equality ::平等财产

Multi-Step Equations ::多级

Equations with Fractions ::带有分数的平方

Examples ::实例

Example 2 ::例2

Review ::回顾

Review ( Answers) ::回顾(答复)

Introduction
::导言

Solving Equations
::溶解等量

One-Step Equations
::单步等量

Properties of Equality
::平等财产

Multi-Step Equations
::多级

Equations with Fractions
::带有分数的平方

Examples
::实例

Example 2
::例2

Review
::回顾

Review ( Answers)
::回顾(答复)