# Linear Equations

A **linear equation **or **first-degree **in is written in standard form as

with

This is solved as follows:

(we subtracted from both sides)

(we divided both sides by )

In many cases, simple first-degree equations can be solved mentally.

**For example**, the solution of is and the solution of is

__Example__:

Solve the linear equation

__Solution__:

We have

(we subtracted from both sides)

(we combined like terms)

(we subtracted from both sides)

(we multiplied both sides by )

(we divided both sides by )

To guard against errors in arithmetics or algebra, it’s a good idea to check the solution by substituting it back into the original equation. Thus, if we substitute in the equation , we obtain , . This shows that is indeed the solution.

__Example__:

Solve the linear equation

__Solution__:

Multiplying both sides of the equation by the LCD and simplifying, we have

That is, or

Adding to both sides of the last equation, we obtain

; that is

From which it follows that . We now check by substituting into the original equation to obtain

an equation in which neither side is defined because of the zeros in the denominators. In other words, the substitution doesn’t make the equation true, it makes the equation meaningless. Here, the correct conclusion is that the original equation **has no root**.