To determine whether the system of linear equations has one solution, infinitely many solutions, or no solution, we can use the following approach:

1. First, write the given system of linear equations clearly:

5x - 6y = 6 \tag{1} 10x - 12y = 12 \tag{2}

2. Now, we will simplify and compare the equations. Notice that the second equation is a multiple of the first equation. Specifically:

$10x - 12y = 2 \cdot (5x - 6y)$

Thus, equation (2) is essentially twice equation (1). Dividing equation (2) by 2 gives us:

$5x - 6y = 6$

This is the same as equation (1), meaning the two equations are dependent and represent the same line.

Conclusion:

Since the two equations represent the same line, the system has infinitely many solutions.

---

Would you like further details or have any questions?

Here are 5 related questions you might want to explore:

1. How can you identify parallel lines in a system of linear equations?
2. What are the conditions for a system of equations to have no solution?
3. How do you solve a system of linear equations using the substitution method?
4. What is the geometric interpretation of infinitely many solutions?
5. How does the determinant of a coefficient matrix affect the number of solutions?

Tip: For any system of two linear equations, if the equations are proportional, the system has infinitely many solutions or no solution, depending on whether the constants are also proportional.