To determine the number of solutions for the system of equations:
[tex]\[
\begin{cases}
6x - y = 5 \\
12x - 2y = 10
\end{cases}
\][/tex]
we need to analyze the relationship between the two equations.
First, observe the equations:
1. [tex]\(6x - y = 5\)[/tex]
2. [tex]\(12x - 2y = 10\)[/tex]
We can rewrite the second equation to see if it is a multiple of the first equation. Divide the second equation by 2:
[tex]\[
12x - 2y = 10 \implies 6x - y = 5
\][/tex]
This shows that the second equation is effectively the same as the first equation after dividing by 2.
Since both equations are essentially the same, they are dependent, which means one is just a multiple of the other. In such cases, instead of having a unique solution, the system has infinitely many solutions (any [tex]\((x, y)\)[/tex] that satisfies the first equation will also satisfy the second).
### Conclusion:
The system of equations has infinitely many solutions.
