algebra, a linear equation is one that contains simply the variable, which makes them
one of the simplest types of equations. For example, a linear equation in one variable
has one unknown (the variable) represented by a letter; this letter, usually x,is always
to the power of 1, meaning there is no x^2 or x^3 in the equation.For instance, x 3 9 is a simple linear equation. To solve such an equation,
one must either add, subtract, multiply, and/or divide both sides of the equation by
numbers and variables—and do this in the correct order—to end up with a solution: a
single variable and single number on opposite sides of the equals sign. In this case, the
solution to the linear equation is x6.Finally, linear equations can be further broken down. For example, in the linear
equation axbyczdwh, in which a, b, c,and dare known numbers and x, y,
z,and ware unknown numbers, if h0, the linear equation is said to be homoge-
neous.What is the absolute valueof a number?
The absolute value of a real number is the number stripped of any negative value.
Therefore, the absolute value of a number will always be greater than or equal to zero.
(Formally, the absolute value is considered the distance of a number from zero on a
number line.) The symbol for “absolute value” is the number inside two parallel verti-
cal lines (| |). For example, the absolute value of xis given as |x|. If the number is nega-
tive within the absolute value sign, it will automatically become positive. In numerical
form, | 3| equals 3 and |3| equals 3.
When discussing complex numbers, the absolute value often means squaring the
numbers, then taking the square root of those numbers. For example, the common way of
writing complex equations is zabi; the absolute value of zbecomes |z| a 2 b 2.
140 For instance, if z^3 ^4 i, then |z| ^32 ^42 , then |z| 5.
What is a diophantine equation?
T
he first mention of diophantine equations was by Greek (Hellenic) mathe-
matician Diophantus (c. 210–c. 290 CE). In his treatise Arithmetica,he solved
equations with several variables for integral solutions—what we call diophantine
equations today. (For more about Diophantus in history, see “History of Mathe-
matics.”) These are represented by one equation with at least two variables, such
as xand y,and whose solutions have to be whole numbers (or integers). These
equations either have no solutions, or an infinite or finite number of solutions.
Diophantine analysisis the mathematical term for how to determine integer
solutions for such algebraic equations.