Problem Solving

AAPL wrote:Source: Official Guide

Which of the following lines in the xy-plane does not contain any points with integers as both coordinates?

$$\text{A. }\ y=x$$
$$\text{B. }\ y=x+\frac{1}{2}$$
$$\text{C. }\ Y=x+5$$
$$\text{D. }\ y=\frac{1}{2}x$$
$$\text{E. }\ y=\frac{1}{2}x + 5$$

The OA is B.

Core concept: If a point (x,y) is on a given line, then the coordinates (x and y) of that point must satisfy the equation of that line.

In answer choice B, the equation (y = x + 1/2) tells us that, for ANY POINT ON THE LINE, the y-coordinate will be equal to the x-coordinate plus 1/2
So, if the x-coordinate is an integer, the y-coordinate cannot be an integer.
So, it's impossible for any point on this line to have integers as BOTH coordinates.

Answer: B

Cheers,
Brent

AAPL wrote:Source: Official Guide

Which of the following lines in the xy-plane does not contain any points with integers as both coordinates?

$$\text{A. }\ y=x$$
$$\text{B. }\ y=x+\frac{1}{2}$$
$$\text{C. }\ Y=x+5$$
$$\text{D. }\ y=\frac{1}{2}x$$
$$\text{E. }\ y=\frac{1}{2}x + 5$$

The OA is B.

Since a non-integer fraction + integer will NEVER = integer, we see that in the equation y = x + 1/2, it's impossible for both x and y to be integers.

Answer: B