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.
\[\begin{array}{rcl} 4 x &=& -12\\
&&\phantom{xxx}\color{blue}{-12\text{ added to each side}}\\
x &=&\dfrac{ -12}{4}\\
&&\phantom{xxx}\color{blue}{\text{both sides divided by }4}\\
x &=& -3\\
&&\phantom{xxx}\color{blue}{\text{simplified}}
\end{array}\]
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