If x and y are ______ then the linear equation 4x+8y = 12 has unique solution.

A) Negative Integers
B) Natural Numbers
C) Real Numbers
D) Rational Numbers

Explain in detail

clearly, (1,1) is a solution

To make it the only one, B needs to be the answer
Otherwise, the solution set lies on a straight line, with many values.
Plot the line and this is easy to se.
Natural numbers lie only in Quadrant I, and exclude the axes.

To determine which values of x and y will result in a unique solution for the linear equation 4x + 8y = 12, we need to consider the nature of the equation itself.

First, let's rewrite the equation in slope-intercept form (y = mx + b):

4x + 8y = 12
8y = -4x + 12
y = (-4/8)x + (12/8)
y = (-1/2)x + (3/2)

In this form, we can see that the coefficient of x is (-1/2), which represents the slope of the line. Since the slope is not zero, the equation represents a non-horizontal line. Additionally, the presence of y in the equation indicates that it is a linear equation, not a constant one.

In order for a system of two linear equations to have a unique solution (a single point of intersection), we need the lines to be non-parallel. In the case of the given equation, the line represented by y = (-1/2)x + (3/2) is non-parallel to any other line as long as its slope is different from the slope of that other line.

Considering these observations, the answer to the question is C) Real Numbers. Any real values of x and y that you plug into the equation 4x + 8y = 12 will result in a unique solution, since the equation represents a non-parallel line in the x-y coordinate plane.