Final answer:
The given system of linear equations is consistent and has one unique solution because the two lines represented by the equations have different slopes, indicating that they intersect at a single point.
Step-by-step explanation:
The question is asking whether the given system of linear equations is consistent or inconsistent. The given system is:
As a first step, we can try to find a solution that satisfies both equations simultaneously. We notice that both equations have the term 3y. Let's compare the right-hand sides of both equations:
- For the first equation, if we divide by 3, we get y = 3x - 2
- For the second equation, after dividing by 3, we get y = 2x - 2
These two equations represent two different lines with different slopes. The first line has a slope of 3, and the second line has a slope of 2. Since the slopes are not equal, the lines are not parallel, and they intersect at a point. This means the system of equations has a single solution where these two lines meet, making it a consistent system.
A system's consistency can also be judged by analyzing if the equations are multiple versions of the same line (same slope and y-intercept), in which case the system would have infinitely many solutions, or if the lines are parallel (same slope, different y-intercept), in which case there would be no solution and the system would be inconsistent.
In this case, since the slopes are different, the lines are not parallel, and thus, there is one unique solution to the system.