What about this?

Question

The graph shows a coordinate plane that goes from negative 5 to 5 in each direction. The graph also contains two lines, each with positive slope. (The slope of each line is 2.) It appears that the lines will never intersect.
Does this system of equations have a solution?

Answer 1

In the context of a system of equations represented by the two lines you described, the situation can be analyzed as follows:

1. Equations of the Lines: Since both lines have a positive slope of 2, their equations can be generally expressed in the form \( y = 2x + b_1 \) and \( y = 2x + b_2 \), where \( b_1 \) and \( b_2 \) are different y-intercepts.

2. Parallel Lines: If the lines have the same slope (which is 2 in this case) but different y-intercepts (meaning \( b_1 \neq b_2 \)), the lines are parallel. Parallel lines, by definition, do not intersect at any point.

3. No Solution: Since the lines will never meet, there is no point \((x, y)\) that satisfies both equations simultaneously. Therefore, the system of equations represented by these two parallel lines does not have a solution.

In conclusion, the system of equations formed by two lines with the same slope but different intercepts has no solution.