Linear equation

A linear equation is an equation where the unknown $ \large x $ only appears in the first power, that is $ \large x^1 $. This means there must not be $ \large x^2 $, $ \large x^3 $ or higher powers of $ \large x $, and $ \large x $ must also not appear in a denominator.

A linear equation can always be written or rewritten in this form:

$$ \large ax + b = 0 $$

When you see $ \large ax $, it actually means $ \large a \cdot x $.

$$ \large ax = a \cdot x $$

Your equation will not always look exactly like this when you get an exercise. It might look like this:

$$ \large 8 + 2x = 16 $$

Or like this:

$$ \large 2(x + 4) = 10 $$

Both are linear equations, because they can be rewritten into the form $ \large ax + b = 0 $.

Example of a solution

If we take the equation:

$$ \large 2(x + 10) = 4x $$

First multiply into the parentheses:

$$ \large 2 \cdot x + 2 \cdot 10 = 4 \cdot x \Leftrightarrow $$

$$ \large 2x + 20 = 4x $$

Then isolate $ \large x $:

$$ \large 2x-2x + 20 = 4x-2x \Leftrightarrow $$

$$ \large 20 = 2x $$

Divide both sides by $ \large 2$:

$$ \large \frac{20}{2} = \frac{2x}{2} \Leftrightarrow $$

$$ \large 10 = x $$

The result is $ \large x = 10 $.

Rules of calculation

When solving linear equations, you may use these rules:

You may add the same number to both sides of the equals sign.
You may subtract the same number from both sides.
You may multiply both sides by the same number.
You may divide both sides by the same number (as long as it is not 0).

The important thing is that you always treat both sides the same, so the equality is preserved.

What is not a linear equation?

To tell the difference, it is good to look at examples that are not linear equations:

Here $ \large x^2 $ appears, so it is a quadratic equation:

$$ \large x^2 + 3x = 0 $$

Here $ \large x $ is in the denominator, so it is not a linear equation:

$$ \large \tfrac{1}{x} = 2 $$

How many solutions?

A linear equation normally has one solution, but there are two special situations:

No solution:

$$ \large 2x + 3 = 2x + 5 $$

Reduces to

$$ \large 3 = 5 $$

Which can never be true.

Infinitely many solutions:

$$ \large 2x + 3 = 2x + 3 $$

Reduces to

$$ \large 3 = 3 $$

Which is always true, no matter what $ \large x $ is.

Summary

A linear equation is an equation where $ \large x $ only appears in the first power. They can always be rewritten into the form $ \large ax + b = 0 $. You solve them step by step by isolating $ \large x $ using the calculation rules. Normally there is one solution, but there can also be none or infinitely many.