Carcass wrote:

Which of the following is an equation of a line that doe NOT contain any points in the xy-plane for which both coordinates are integers?

A.\(y=4\)

B. \(y=\frac{1}{2x}\)

C.\(y=x+3\)

D. \(y= x+\frac{1}{2}\)

E. \(y=\frac{1}{2x} + 3\)

The question or OA is flawed..

A.\(y=4\).... x could be any integer...both integer possible

B. \(y=\frac{1}{2x}\).......2xy=1 or xy=\(\frac{1}{2}\)..

If both x and y are integers, xy should be an integer... so both x and y cannot be integers

C.\(y=x+3\)... Possiible

D. \(y= x+\frac{1}{2}\).......\(y- x=\frac{1}{2}\)..

The difference of two integers cannot be a fraction... so both x and y cannot be integers

E. \(y=\frac{1}{2x} + 3....y-3=\frac{1}{2x}\)........If x is an integer, \(\frac{1}{2x}\) will be a fraction, then y-3 cannot be a fraction

so answer = B, D, and E

Note - editing the answer accordingly

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Some useful Theory.

1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048

2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825

3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html

4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html

5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html