How do you solve the following system?: # 3x-12y=4 , x-4y=3 #

Both of these equations are in the Standard Form. Therefore we can use this rule to determine the slope of the line represented by these two equations:

The standard form of a linear equation is: #color(red)(A)x + color(blue)(B)y = color(green)(C)#

Where, if at all possible, #color(red)(A)#, #color(blue)(B)#, and #color(green)(C)#are integers, and A is non-negative, and, A, B, and C have no common factors other than 1
The slope of an equation in standard form is: #m = -color(red)(A)/color(blue)(B)#

Slope for equation 1)

#m = -color(red)(3)/color(blue)(-12) = 1/4#

Slope for equation 2)

#m = -color(red)(1)/color(blue)(-4) = 1/4#

Because both these lines have the same slope by definition they are parallel.

Parallel lines either represent the same lines or they have no points in common and therefore there is no solution or the solution is the null or empty set.

We can multiply both sides of the second equation by #color(red)(3)# to make the left sides of each equation the same:

#color(red)(3)x - (color(red)(3) xx 4y) = color(red)(3) xx 3#

#3x - 12y = 9#

With the left side of both equations the same but the right side of both equations different we know these lines have the same slope but are not the same line.

Therefore there is no solution to this problem or the solution is the null or empty set #{O/}#