Which of the following points is not on the line y = 5x + 3

Ah, there's a difference between 'irrational' and 'non-terminating'. 1/3 = 0.33333.... is 'non-terminating', but it is also certainly a rational number. Rational numbers are those that can be written as fractions using only integers. As decimals, rational numbers can be terminating or non-terminating, but when rational numbers produce non-terminating decimals, they *always* produce repeating (sometimes called recurring) decimals; a certain pattern of digits repeats forever. Irrational numbers are those numbers like \(\Pi\) and \(\sqrt{2}\) which cannot be written as fractions involving integers. As decimals, irrational numbers have no pattern of digits that repeats forever.

Which of the following points is not on the line \(y = 5x + 3\)?

A. \((\frac{1}{2}, \ \frac{11}{2})\)
B. \((\frac{1}{3}, \ \frac{14}{3})\)
C. \((\sqrt{8}, \ 3 + 10*\sqrt{2})\)
D. \((\sqrt{4}, \ 13)\)
E. \((\sqrt{2}, \ \frac{31}{3})\)


Note that for a point \((x, y)\) to lie on the line \(y = 5x + 3\), the coordinates \(x\) and \(y\) must satisfy the equation \(y = 5x + 3\). It is important to recognize that if \(x\) is an irrational number, then \(5x + 3\) will also be irrational, and as a result, \(y\) must be an irrational number as well. Conversely, if \(x\) is a rational number, then \(y\) must also be a rational number. Among the given options, only option E, \((\sqrt{2}, \ \frac{31}{3})\), violates this rule: \(x\) is irrational while \(y\) is rational. Therefore, for option E \((y = \text{rational}) \neq (5x + 3 = \text{irrational})\), which means that the point \((\sqrt{2}, \ \frac{31}{3})\) does not lie on the line \(y = 5x + 3\).


Answer: E