Final answer:
The given system of equations has infinitely many solutions because the two equations represent the same line. The solutions can be expressed as ordered pairs in the form (x, (3/2)x - 4), where x is any real number.
Step-by-step explanation:
To solve the system of equations, we must find the value of the variables x and y that satisfy both equations simultaneously. The given equations are:
We notice that the second equation is exactly -2 times the first equation. This means the two equations are proportional and represent the same line. Therefore, there are infinitely many solutions to this system, and it does not have a unique solution. The solution set can be expressed in terms of one variable, for example y, as any point along the line represented by either equation. Hence, the system is dependent and consistent.
To express the solutions as ordered pairs, we can solve the first equation for y:
- 3x - 2y = 8
- 2y = 3x - 8
- y = (3/2)x - 4
Therefore, the solutions can be written as (x, (3/2)x - 4), where x can be any real number.
We should always examine our answer to ensure it is reasonable, both algebraically and graphically. Since the two equations are consistent and dependent, our solution set is reasonable and reflects the fact that the equations describe the same line.