Chapter 2 Equations in one Variable

Section 2.1 Simple Equations

2.1.4 Solving linear Equations

Info 2.1.14

A linear equation is an equation in which only multiples of variables and constants occur.
For a linear equation in one variable (here the variable

x

) one of the following three statements holds:

The equation has no solution.
The equation has a single solution.
Every value of $x$ is a solution of the equation.

These three cases are distinguished by means of the transformation steps:

If the transformation ends up in a statement that is wrong for all $x$ (e.g. $1 = 0$ ) then the equation is unsolvable.
If the transformation ends up in a statement that is true for all $x$ (e.g. $1 = 1$ ) then the equation is solvable for all values of $x$ .
Otherwise, the equation can be solved, i.e. it can be transformed into the equation $x = value$ which is the solution.

Set notation 2.1.15

Using set notation (with

L

as the conventional symbol for the solution set) these cases can be expressed as follows:

$L = {}$ or $L = ∅$ if there is no solution,
$L = {value}$ if there is a single solution,
$L = ℝ$ if all real numbers $x$ are a solution.

Example 2.1.16

The linear equation

3 x + 2 = 2 x - 1

has one solution. This solution is obtained by equivalent transformations:

3 x + 2 = 2 x - 1 \underset{- 2 x}{\underset{⏟}{\Leftrightarrow}} x + 2 = - 1 \underset{- 2}{\underset{⏟}{\Leftrightarrow}} x = - 3 .

Hence,

x = - 3

is the only solution.

Example 2.1.17

The linear equation

3 x + 3 = 9 x + 9

has the solution:

3 x + 3 = 9 x + 9 \underset{: (x + 1)}{\underset{⏟}{\Leftrightarrow}} 3 = 9 .

This statement is wrong. Hence, for all

x \neq - 1

(transformation condition) the equation is wrong. Inserting

x = - 1

satisfies the equation, and so the only solution is

x = - 1

.
Alternatively, the equation could have been transformed as follows:

3 x + 3 = 9 x + 9 \underset{- 3 x - 9}{\underset{⏟}{\Leftrightarrow}} - 6 = 6 x \Leftrightarrow x = - 1 .

Exercise 2.1.18

Transform the following linear equations and specify their solution sets: Enter simply ${a}$ for a unit set and

{}

for an empty set.

The equation $x - 1 = 1 - x$ has the solution set $L$ $=$ ,
The equation $4 x - 2 = 2 x + 2$ has the solution set $L$ $=$ ,
The equation $2 x - 6 = 2 x - 10$ has the solution set $L$ $=$ .

The first equation can be transformed into

2 x = 2

x = 1

, respectively, so the solution set is

L = {1}

. The second equation can be transformed into

2 x = 4

and the solution set is

L = {2}

. The third equation can be transformed into

- 6 = - 10

which is a false statement, hence

L = {}

Exercise 2.1.19

Find the solution of the general linear equation

a x = b

with

a

and

b

being real numbers. Specify the values of

a

and

b

for which the following three cases occur:

Every value of $x$ is a solution ( $L = ℝ$ ) if $a$ $=$
and $b = 0$ .
There is no solution ( $L = ∅$ ) if $a$ $=$
and $b \neq$ .
Otherwise, there is a single solution, namely $x$ $=$ .

Every value of

x

is a solution (

L = ℝ

) if

a = 0

and

b = 0

. There is no solution (

L = ∅

) if

a = 0

and

b \neq 0

. Otherwise, there is only one solution, namely

x = \frac{b}{a}

Onlinebrückenkurs Mathematik

1. Elementary Arithmetic

2. Equations in one Variable

3. Inequalities in one Variable

4. System of Linear Equations

5. Geometry

6. Elementary Functions

7. Differential Calculus

8. Integral Calculus

9. Objects in the Two-Dimensional Coordinate System

10. Basic Concepts of Descriptive Vector Geometry

11. Language of Descriptive Statistics