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= 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  =  2x-1     -2x     x+2  =  -1     -2     x  =  -3.

Hence, x=-3 is the only solution.

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

3x+3  =  9x+9     :(x+1)     3  =  9.

This statement is wrong. Hence, for all x-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  =  9x+9     -3x-9     -6  =  6x        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.  
  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=) if a =
    and b=0.
  • There is no solution ( L=) if a =
    and b .
  • Otherwise, there is a single solution, namely x = .