Bragg's equation has no solution, though

Special cases when solving equations

When solving an equation, these special cases can occur:

  1. As a solution set are all rational numbers possible. $$ L = {QQ} $$

  2. The equation is at no inserted number correct. $$ L = {$$ $$} $$

  3. 0 is the solution to the equation. $$ L = {0} $$

1. All rational numbers are possible as a solution set. $$ L = {QQ} $$

Example:

$$ 2 * x + 2 = 2 * x + 2 $$ You remove two $$ x $$ boxes.

$$2=2$$

It creates a true statement on the last line of the solved equation.

You can now put any weight you want in the $$ x $$ box. Since you're doing it on both sides of the scale, the scales will hang in balance.

Write down the solution set as follows: $$ L = {QQ} $$

is the unknown weight box.

stands for 1 kg.

If you do another equivalent conversion, you get $$ 0 = 0 $$.

2. The equation is not correct for any number inserted. $$ L = {$$ $$} $$

Example:

$$ 2 * x + 2 = 2 * x + 4 $$ You remove two $$ x $$ boxes.

$$ 2 = 4 $$ That's one wrong statement.

The equation is not solvable. That is, the solution set is empty.

Write down the solution set as follows: $$ L = {$$ $$} $$

Summary of the two special cases:

Whenever, in the equation, the $$ x $$ceases to exist, has the equation

  • either no Solution $$ L = {$$ $$} $$
  • or infinitely many Solutions $$ L = {QQ} $$.

The scale model is crossed out because the scale out of balance hangs.

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3. 0 is the solution to the equation.
$$ L = {0} $$

Example:

$$ 5 * x = 7 * x $$ $$ | -7 * x $$

$$ - 2 * x = 0 $$ $$ |: (- 2) $$

$$ x = 0 $$

$$ L = {0} $$

If each $$ x $$ box weighs $$ 0 $$ kg, the scales are in equilibrium.

This reshaping is not permitted:

$$ 5 x = 7 x $$ $$ |: x $$

$$5=7$$

Here you would assume that $$ x $$ is not $$ 0 $$, because you cannot divide by 0. The 0 is just the solution.