一元一次方程式 (One-Variable Linear Equation)

What can a linear equation be used in our actual life? Well, it could be very useful in solving a problem by transforming it into an equation, for instance:

Lily's grandmom's age is 7 times of Lily's. 9 years later, her grandmom's age is 4 times of her age.
How old is Lily and her grandmom now?

 Solution: 

let:

Lily is x years old,

her grandmother is 7x years old.

9 years later,

Lily is x+9 years old,

Her grandmother has an age of 7x+9.

since, grandmother's age is 4 times of Lily's age,

we can arrange the equation to:

4(x + 9 ) = 7x + 9 

solve x to get what is Lily's age:

NOTE: A(B + C) = AB + AC, WE GET:

4x + 36 = 7x + 9

7x - 4x = 36 - 9

3x = 27

x = 9

So, Lily is 9 years old, her grandmother is 63 years old.

As we can see from the example above, it is useful!
Before we go through the application of this one-variable linear equation,
Let's do some practice:

Solve x :

1. 5(x - 5) = 3(x - 1)

Solution: 

NOTE: A(B - C) = AB + AC, WE GET:

5x - 25 = 3x - 3    OR (-3 + 3x)
5x - 25 - 3x = -3
5x - 3x = -3 + 25 OR (25 - 3)
2x = 22
x = 22 / 2 = 11

NOTE THAT, 3x from RIGHT HAND SIDE MOVE TO LEFT HAND SIDE WILL GET A -3x
(FROM +3x to -3x),
while -25 FROM LEFT HAND SIDE MOVED TO RIGHT HAND SIDE WILL GET A +25

2.   x / 15 + x / 21 + x / 12 = 1

Solution: 

3. 

Solution: 

Order of Operation, How to simplify and solve a complex Addition, Subtraction, Multiplication Problem

Problem: 3 + 4 x 2 = ?

Let's look at the question shown above, two students solve the problem with different result:

Student 1                                Student 2

3 + 4 x 2 =                            3 + 4 x 2 =

7 x 2 = 14                             3 + 8 = 11

It seems that each student interpreted the problem differently, resulting in two different answers. 

When performing arithmetic operations there can be only one correction answer. We need a set of rules in order to avoid this kid of confusion. Mathematicians have devised a standard order of operations for calculations involving more than one arithmetic operation:

Rule 1:  First, perform any calculations inside parentheses. (parentheses : (), [], {}, <>)

Rule 2:  Next perform all multiplications and divisions, working from left to right.

Rule 3:  Lastly, perform all additions and subtractions, working from left to right.

10105 x 97 +99 x 105 = ?

Have you ever tried to solve the question shown above without going through a complex multiplication? Here is the way how:

10105 x 97 + 99 x 105 = (10000 + 100 + 5) x 97 + (100 - 1）x 105

                                 = 970000 + 9700 + 97 x 5 + 10500 - 105

                                 = (970000 + 9700) + (100 - 3) x 5 + (10500-105)

                                 = 979700 + 500 - 15 + 10395

                                 = (979700 + 500) + (10395 - 15)

                                 = 980200 + 10380

                                 = 990580

To simplify and solve the problem shown above, We need  to remember the following equations and concepts show as below: 

Please be note that, when we are solving a problem with +, -, x and /, without any bracket in the question, 

A x B = AB      (For instance, 4 x 5 = 4(5) = 20)

For most of the problems, the multiplication of two numbers are expressed in the way shown above. 

A(B+C）= AB + AC   (For instance, 4(5) = 4(2 + 3) = 4 x 2 + 4 x 3 = 8 + 12 = 20)

A(B - C) = AB - AC    (For instance, 4(9) = 4(10-1) = 4 x 10 - 4 x 1 = 40 - 4 = 36)

-A (B - C) = -AB + AC   (For instance -5(-2) = -5(1 - 3) = -5 x 1 + -5(-3) = -5 + 15 = 10 )

A (B+C-D) = AB + AC -AD

PLEASE BE REMINDED THAT 

BOTH POSITIVE (+) MULTIPLIES TOGETHER WILL GET A POSITIVE (+) 

BOTH NEGATIVE (-) MULTIPLIES TOGETHER WILL GET A POSITIVE (+)

A NEGATIVE (-) MULTIPLIES WITH A POSITIVE (+) WILL GET A NEGATIVE (-)

Let's go for a more complicated question：

101 x 99 - 102 x 98 + 103 x 97 - 104 x 96

= (100 + 1) x 99 - (100 + 2) x 98 + (100 + 3) x 97 - (100 + 4) x 96

= [99(100) + 99] - [100(98) + 2(98)] + [100(97) + 3(97)] - [(100)(96) + 4(96)]

= (9900 + 99) - [9800 + 2(100-2)] + [9700 + 3(100 - 3)] - [9600 + 4(100-4)]

= (9900 - 9800 + 9700 - 9600) + 99 - [2(100) - 2(2)] + [3(100) - 3(3)] - [4(100) - 4(4)]

= 200 + 99 -(200-4) + (300-9) - (400 - 16)

= 200 + 99 -200 + 4 +300 - 9 - 400 + 16

= (200 - 200 + 300 - 400) + (99 + 4 - 9 + 16)

= -100 + 110

= 10

素数(Prime Number), 最小公倍数 ( Lowest Common Multiple) 和 异分母分数加减法 (addition & subtraction of fractions)

相信 分数 的 加减法在小学的课程已经有 带过
而我还记得 小学 老师 是 用 图画 来表示 分数 让同学更容易吸收和明白

以下 简单的 例子 在小学 应该都学过：

以上例子显示 不同分母 的分数 在相加 之前 都要 讲分母 同化，

而最简单的例子就是 讲两个不同的分母相乘 得到一个 相同的倍数 (Common Multiple Number)

另外一个 简单的例子 就是 两个相加的 分数，

其中一个分母 是另外一个分母的倍数 : 

以上的两个例子都是简单的分数加减题目。
那如果有超过三个或以上不同的分母 的加法呢？

1/18+1/45+1/12 = ？

如果为了得到同样的分母而将三个分母乘起来 分母会变得很大
解题就会变得困难许多了

当遇到这样的题目，就必须先找到三个分母的最小公倍数 （lowest commo multiple number)
获得最小公倍数的方法如下:

1) 若分母不是素数 （Prime Number）『只能被自己或者1除的号码 example: 2,3,5,7,11,13,17,23...』可以将其分解成由素数相乘而得到的号码， 例如：

45 = 3 x 3 x 5,
12 = 2 x 2 x 3,
18 = 2 x 3 x 3

2) 将所有分母分解成由素数组成的号码后 的所有素数例出来：

2，3，5

3）分析 每个素数在不同分母 分解后出现过 最多次 的 次数:

例如 2 在三组分母里面 出现次数最多为2 次 （45）
         3 在三组分母出现次数最多为 2 次 （45 和18）
         5 在三组分母出现次数最多为 1 次 （45)

4）最小公倍数 = 2 x 2 x 3 x 3 x 5 = 180    (2 和 3 出现两次，所以2 和 3 乘两次 ）

得到 最小公倍数 之后 就能够将所有 分母通分啦～:

再po 多一个简单的例子 让同学更好的理解：

1)49 = 7 x 7

21 = 7 x 3

28 = 7 x 2 x 2

2) 素数 : 2,3,7

3) 7 在 个别三组出现最多为 2 次

    2 在个别组合出现最多为 2 次

    3 在个别组合出现最多为 1 次

4) LCM = 7 x 7 x 2 x 2 x 3

就这么简单哦
同学必须做多几次，

才能快速掌握 解答的技巧 =)

学习愉快～！