Final answer:
The system of equations 4x + 10y = 20 and 6x + 15y = 30 simplifies to the same single line, which means the system has an infinite number of solutions because the equations represent the same line.
Step-by-step explanation:
The question involves determining the number of solutions a system of equations has. The given system is:
- 4x + 10y = 20
- 6x + 15y = 30
To solve this, we'll first simplify each equation to its lowest terms:
- Divide the first equation by 2: 2x + 5y = 10
- Divide the second equation by 3: 2x + 5y = 10
As we can see, after simplification, both equations are identical, which means they represent the same line. Therefore, this system of equations has an infinite number of solutions, since any point that lies on one line will also lie on the other.
In conclusion, because the equations are identical after simplification, the system's equations are dependent and have an infinite number of solutions.