Defend the Justice of the Equals Sign. What Linear Equations Teach Us About Balancing Life [A Super Introduction to Mathematics for the Third-Class Chief Electrical Engineer Examination]

Hello everyone! I received a huge response to my previous explanation of "addition and subtraction."

Thank you, truly, from the bottom of my heart.

Today, we are tackling the major hurdle of linear equations: "multiplication and division"... and that dreaded "fraction." Many of you likely have bitter memories, such as "I wanted to close my textbook the moment I saw a fraction" or "I always get confused about which number to divide by."

But don't worry! You aren't "bad at fractions." You just weren't taught that a fraction is a "translation of division"! From this moment on, your perspective on mathematics will change completely. Are you ready!

1. A "fraction" was just "division lined up vertically"!

First, let's expose the true nature of this form that makes everyone go "ugh."

$$
\frac{x}{5}
$$

This "horizontal bar." It is actually a relative of the division symbol "$${÷}$$". There is only one rule for translating fractions.

• The top number (numerator) is the "dividend (left side)"!

• The bottom number (denominator) is the "divisor (right side)"!

In other words, "top is left, bottom is right"!

How about that! It's not scary anymore, is it? A fraction is nothing more than "division standing up on its tiptoes"!

2. Transposing division (fractions): The passion to blow away the denominator!

Now, let's actually solve a problem. I will provide a play-by-play without taking a single step back or skipping a single line!

$$
\frac{x}{4}=6
$$

【Step 1】 Translate and reveal its true form!

First, rewrite the fraction using the "top is left, bottom is right" rule.

$$
x÷4=6
$$

[Step 2] Eliminate the annoying "$${÷4}$$"!

Our goal is to reach "$${x=…}$$". Right now, the calculation "divide by 4" is sitting right next to x. How do we get rid of it...? That's right! We just need to use the opposite calculation: "multiply by $${4}$$"!

[Step 3] Honestly multiply both the left and right sides by "$${4}$$"!

To keep the scale balanced, we multiply both the left and right sides by 4 fairly!

$$
(x÷4)×4=6×4
$$

[Step 4] The truth is revealed!

On the left side, since we "divide by $${4}$$ and multiply by $${4}$$", they cancel each other out, leaving only x.

$$
x=6×4
$$

$$
x=24
$$

Did you see it! The "$${4}$$ (the divisor on the right)" that was under the fraction crossed the bridge and turned into "multiplication" when it went to the right side. This is the truth about transposing division!

3. Transposing Multiplication: A Step Toward Freedom!

Finally, there is multiplication.

$$
3x=15
$$

"$${3}$$ times $${x}$$ equals $${15}$$". You already know how to get rid of the "multiplication by $${3}$$" next to $${x}$$. We "divide by $${3}$$"!

[A play-by-play account without skipping a single line!]
Divide both the left and right sides by 3 fairly!

$${\frac{3x}{3} =\frac{15}{3}}$$

On the left side, $${3}$$ divided by $${3}$$ becomes "$${1}$$". In other words, only "$${x}$$" remains!

$${1x=5}$$

You have finally reached the answer!

$${x=5}$$

The "multiplication" that was on the left has transformed into "division (denominator)" on the right!

A moment of "self-renewal" that turns the past into inspiration!

Transposition is an honest story of treating the left and right sides fairly, and the "I don't understand" of the past was the best preparation period for deep understanding today.
Having obtained the key that fractions mean "top is left, bottom is right," you have certainly surpassed the you of yesterday and are now moving forward powerfully.

For you who live life to the fullest and can relearn from any point, there is no longer any mathematical door you cannot open. Praise yourself for taking that first step, and let's move forward together toward an exciting future where infinite possibilities unfold.

What changes life in a big way is always the determination made from this moment on.Come on, if you don't do it now, when will you?

See you next time!

How is your preparation for the Second-Class Electrician written exam going?

How about a little break from your studies?