If we consider each side of the equation separately we see that they each represent straight lines in the form y=mx+b (the slope-intercept form). The equation y=3x+14 is a line with slope of 3 and y-intercept of 14. The equation y=3x is a line with slope of 3 and y-intercept of 0. These lines have the same slope so they must be parallel, and therefore they have no points in common. In other words, they never cross! There is no value of x that satisfies the equation 3x+14=3x. For short, we can just say "no solution."

If we change the equation slightly so that 3x+14=3x+14 we get a very different answer. If we subtract 3x from both sides we get 14=14 which is always true. We could also subtract 14 from both sides then divide both sides by 3 and we get x=x, which is also always true. Using the graphical approach it's easy to see that each side of the equation is the SAME line. This means the two lines have ALL points in common and therefore EVERY value of x satisfies the equation!

A solution would be a value of x for which the equation is true. Since subtracting 3x from both sides gives the equation 14 = 0 which is false, there is no solution.

one can prove that there is no solution by 

Contradiction. If there is a solution we get 14=0 after simplification

Which is false. So we conclude that there is no solution.