Final Answer:
The given system of linear equations, y = 6x + 2 and y = 6x + 27, does not have a solution.
Step-by-step explanation:
To determine whether a system of linear equations has a solution, we need to examine the coefficients and constants. In this case, both equations have the same slope (6), indicating parallel lines. Parallel lines never intersect, meaning there is no point of intersection, and therefore, no solution exists for the system.
The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. In both given equations, the slope (6) is the same, but the y-intercepts (2 and 27) are different. When two lines have the same slope and different y-intercepts, they are parallel and will never intersect.
Mathematically, we can represent the parallel lines as y = 6x + b₁ and y = 6x + b₂. Since b₁ and b₂ are not equal in this case (2 and 27), the lines will not intersect. This lack of intersection points means there is no common solution for the system.
Understanding the geometric interpretation of parallel lines helps in quickly determining whether a system of linear equations has a solution or not. In this scenario, the parallel lines prevent the existence of a solution for the given system.