Linear Equations with a Single Unknown: Notion
The notion of linear equation
Let #x# be a variable.
A linear equation with unknown #x# is an equation of the form \[ax+b=0\tiny,\] where #a# and #b# are real numbers.
Solving this equation means finding all values of #x# for which the equation is true. Such a value is called a solution to the equation. All values of #x# for which the equation is true, form the solution to the equation.
Equations with unknown #x# are also called equations in #x#.
The expression to the left of the equality sign (#=#) is called the left hand side of the equation (in the above equation, this is #ax+b#) and the expression to the right of the equality sign is called the right hand side (in the above equation, this is #0#).
The expressions #ax# and #b# on the left hand side are called terms. As #b# and #0# occur without #x#, these are called constant terms, or just constants. The number #a# is the coefficient of #x#.
For those who know what a function is: solving the equation determines the values #x# where the linear function #ax+b# assumes the value #0#.
Linear equations are also called first degree equations as the highest power of the unknown #x# is at most #1#. The term first degree is derived from the theory of polynomials.
Of the above definitions, terms and constants have been introduced before.
In this chapter we deal with linear equations with a single unknown.
For #a=2# and #b=3# the equation is #2x+3 = 0#, and #x = -1 \frac{1}{2}# is a solution.
Actually, it is the only solution (there are no others).
We say that #x= -\dfrac{3}{2}# is the solution to the equation #2x+3 = 0#.
All solutions from an equation taken together form the solution set.
The solution set can also be indicated with #x= -1 \frac{1}{2}#.
Here are some examples of equations that can be reduced to linear equations.
\[\begin{array}{lll}2x+3=5x-6 &\quad\quad\quad& \dfrac{2x-3}{6x-5} =3 \\ \dfrac{2}{x+3}+3=\dfrac{5}{x+3}-6& \quad\quad\quad& |2x+3|=|5x-6| \end{array}\]
The first example, #2x+3=5x-6#, is very close to the linear equation as defined above: by carrying all terms to the left and collecting them, we can rewrite it to a real linear equation: #-3x+9=0#. Therefore, it is often also called a linear equation.
In this chapter, we will deal with all of these types of equation.
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