#### 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 $x=\text{value}$ 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:
• $L=\left\{\right\}$ or $L=\text{∅}$ if there is no solution,
• $L=\left\{\text{value}\right\}$ if there is a single solution,
• $L=ℝ$ if all real numbers $x$ are a solution.

##### Example 2.1.16
The linear equation $3x+2=2x-1$ has one solution. This solution is obtained by equivalent transformations:

$3x+2\mathrm{ }=\mathrm{ }2x-1\mathrm{ }\mathrm{ }\underset{-2x}{\underset{⏟}{⇔}}\mathrm{ }\mathrm{ }x+2\mathrm{ }=\mathrm{ }-1\mathrm{ }\mathrm{ }\underset{-2}{\underset{⏟}{⇔}}\mathrm{ }\mathrm{ }x\mathrm{ }=\mathrm{ }-3 .$

Hence, $x=-3$ is the only solution.

##### Example 2.1.17
The linear equation $3x+3=9x+9$ has the solution:

$3x+3\mathrm{ }=\mathrm{ }9x+9\mathrm{ }\mathrm{ }\underset{:\left(x+1\right)}{\underset{⏟}{⇔}}\mathrm{ }\mathrm{ }3\mathrm{ }=\mathrm{ }9 .$

This statement is wrong. Hence, for all $x\ne -1$ (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:

$3x+3\mathrm{ }=\mathrm{ }9x+9\mathrm{ }\mathrm{ }\underset{-3x-9}{\underset{⏟}{⇔}}\mathrm{ }\mathrm{ }-6\mathrm{ }=\mathrm{ }6x\mathrm{ }\mathrm{ }⇔\mathrm{ }\mathrm{ }x\mathrm{ }=\mathrm{ }-1 .$

##### Exercise 2.1.18
Transform the following linear equations and specify their solution sets: Enter simply $\left\{a\right\}$ for a unit set and $\left\{\right\}$ for an empty set.
1. The equation $x-1=1-x$ has the solution set $L$$=$ ,
2. The equation $4x-2=2x+2$ has the solution set $L$$=$ ,
3. The equation $2x-6=2x-10$ has the solution set $L$$=$ .

##### Exercise 2.1.19
Find the solution of the general linear equation $ax=b$ 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$$=$
and $b=0$.
• There is no solution ($L=\text{∅}$) if $a$$=$
and $b\ne$ .
• Otherwise, there is a single solution, namely $x$$=$ .