# How to solve 4x 5 21

Why is?
Formulas are needed to calculate the most varied of sizes.

13 x + 170

The formulas have variables (e.g. x) in which you can insert numbers and then get different results. It is often the case that the result is fixed, but you want to know which number (which x) you have to insert in order to get this result.

13 x + 170 = 400 x = ???

So there is an equation. You know the result and want to get the right x. Equations are puzzles. I'll tell you what will come out, tell me what was used to make it. In this lesson you will learn 1. Solve simple equations. 2. The correct order of transformations. 3. Simplify equations before transforming. 4. How to deal with strange solutions.
Solve simple equations
Solving an equation means transforming it in such a way that you end up with x: x = .....

You have toon both sides perform the same transformation.

As if a retailer used to want to use a scale to find out the contents of a sealed bag. He took away the same number of coins on both sides, added them, halved, doubled the number of bags and coins, etc. But always the same on both sides!

Insanely difficult, isn't it? ;O)    Always perform the counter operation:  The four simplest transformations

x + 6 = 15 | - 6
x = 9 L = {9}

x - 7 = 18 | + 7
x = 25 L = {25}

x 3 = 24 | : 3
x = 8 L = {8}

x: 6 = 7 | · 6
x = 42 L = {42}

The calculation that is carried out is written with a command line after the equation.

At the end you answer the question about the correct x by giving a solution set.

As you can see, you always have to perform the counter-operation. If the equation contains x + 6, you have to calculate - 6 to remove the 6. Just like the dealer with the scales. The order of the transformations
But formulas are often not as simple as just shown. Therefore, you usually need several transformation steps until you get x out.

always
first plus and minus,
then only times and shared! 3 x + 7 = 22 | - 7th
3 x = 15 | : 3
x = 5 L = {5}

Not difficult either! The only difficulty is remembering the correct order:

First plus and minus, then first time and divided!

Why actually? The formula "packs" the x twice: First, the x is multiplied by 3, then 7 is added, because point calculation is more important than line calculation. When "unpacking" we logically have to proceed the other way round: We first undo the +7 through --7. And then the 3 through: 3.

Examples:

4x - 7 = 19 | + 7
4x = 26 | : 4
x = 6.5 L = {6.5}

6 + 2x = 34 | - 6
2x = 28 | : 2
x = 14 L = {14}

x: 6 - 12 = -7 | + 12
x: 6 = 5 | · 6
x = 30 L = {30}
Simplify first, then reshape
Often the terms that make up an equation are unnecessarily complicated. Such equations cannot be solved immediately because it is far too easy to make mistakes in the transformations. No: Such equations are first simplified, only then solved!

2x - 4 + 3x + 2 = 53 (simplify first)
5x - 2 = 53 | + 2
5x = 55 | : 5
x = 11 L = {11} You don't really have to calculate the division at the very end. It is enough to write them down as a fraction. That saves a lot of work! However, the fraction must be completely shortened.
Examples:

-13x + 45 + 6x - 17 = 84
-7x + 28 = 84 | - 28
-7x = 56 | : (-7)
x = -8 L = {-8}

8x + 5 - 9 + 13x = 23 - 8
21x - 4 = 15 | + 4
21x = 19 | : 21
x = L = {} Unusual solutions
There are two situations that lead to a rather unusual solution.

1. Empty set

4x + 6 - 4x +1 = 13 (simplify)
7 = 13

The x-s have been omitted completely. 4x - 4x results in 0x and you don't even write it down. Because you don't brag to friends about the things you don't have: 0 horses, 0 Ferraris and 0 million. :O)

Now there is this strange situation: 7 = 13 And the question arises: Which number can I substitute for x so that 7 equals 13? Answer: None! 7 is never 13. There is no x that solves our equation! Therefore the solution set is empty: L = {}

Empty set:

L = {} or L = ∅

All numbers:

L = Q 2. All numbers

9x - 3 + 2x + 12 - 11x = 9 (simplify)
9 = 9

Again, all of the x's have disappeared. 9 = 9. Which number can I substitute for x so that 9 equals 9? All! 9 is always 9. All numbers solve our equation. All numbers, i.e. the set Q: L = Q

Examples:

4x - 8 - 4x = 9
-8 = 9 L = {}

6x + 12 - 6x = 12
12 = 12 L = Q