Let us solve the given system of equations to identify if it has infinitely many solutions or no solution.

The equations provided are:

1. $5x + 15y = -8$
2. $x = -3y + 2$

Substitution Method:

We substitute $x = -3y + 2$ into the first equation:

$5(-3y + 2) + 15y = -8$

Simplify:

$-15y + 10 + 15y = -8$

Combine like terms:

$10 = -8$

This is a contradictory statement, meaning that the equations do not intersect. Therefore, the system has no solution.

Final Answer:

• The resulting equation is a false statement.
• The system of equations has no solution.

Would you like me to go into further details about this process or related concepts?

Additional Questions:

1. How do we recognize when a system of equations has infinitely many solutions algebraically?
2. What happens geometrically when two equations have no solution?
3. Can substitution always determine the type of solution a system has? Why or why not?
4. What are the steps to use elimination to confirm this result?
5. How would the system change if the second equation were $x = -3y - 2$ instead?

Tip:

Always simplify equations fully before making conclusions about the nature of solutions. Contradictions like $10 = -8$ are a clear sign of no solutions!