### Functions: Lines and linear functions

### Linear equations with a single unknown

Let #x# be a *variable*.

A **linear equation** with **unknown** #x# is an equation that has form \[a\cdot x+b=0,\] in which #a# and #b# are real numbers.

**Solving** the equation is finding all values of #x# for which the equation is true. Such a value is called **a solution** of the equation. The values of #x# for which the equation is true, form **the solution** of the equation, also called the **solution set**.

Equations with #x# as unknown are called equations **in** #x#.

The expression to the left of the equal sign (#=#) is called the **left-hand side** of the equation (for the equation above this is #a\cdot x+b#), and the expression on the right of it is **the right hand side** (for the equation above this is #0#).

The expressions #a\cdot x# and #b# in the left hand side are called **terms.** Because #b# and #0# occur without #x#, they are called **constant terms,** or simply **constants**. The number #a# is called **coefficient** of #x#.

For #a=2# and #b=3# the equation is #2x+3 = 0#, and #x = - \dfrac{3}{2}# is a solution. It is even*, the solution*: there are no other. We say that #x= -\dfrac{3}{2}# is the solution of the equation #2x+3 = 0#. The solution set can also be specified as #\{-\frac{3}{2}\}#.

The type of equation #2x+3=5x-6 # is very close to the real linear equation to: by moving all terms to the left hand side and taking them together, we can rewrite it as a real linear equation #-3x+9=0#. Therefore, this type is also called a **linear equation.** Even more general: if all terms in the equation are constants or constant multiples of #x#, then the equation is called linear.

In terms of a *function* solving the equation is finding all points #x# where the linear function #a\cdot x+b# is equal to #0#.

Instead of *linear* we can also say *of first degree,* because the highest degree in which the unknown #x# occurs is no higher than #1#. The name *of first degree* comes from the theory of *polynomials*.

In this chapter we will first deal with linear equations with a single unknown and *later* we will deal with linear equations with two unknowns.

This answer can be found as follows.

\[\begin{array}{rcl} 5 x &=& 15\\

&&\phantom{xxx}\color{blue}{15\text{ added to each side}}\\

x &=&\dfrac{ 15}{5}\\

&&\phantom{xxx}\color{blue}{\text{both sides divided by }5}\\

x &=& 3\\

&&\phantom{xxx}\color{blue}{\text{simplified}}

\end{array}\]

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