To determine which system has no solution and which has infinitely many solutions, we need to examine the relationships between the equations in each system.

System 1

Equations:

1. $5x + y = 13$
2. $20x + 4y = 64$

Let's analyze these equations:

• Notice that if we multiply the first equation by 4, we get: $4 \times (5x + y) = 4 \times 13$ which simplifies to: $20x + 4y = 52$
• However, the second equation in System 1 is $20x + 4y = 64$, which is different from $20x + 4y = 52$.
• Since $20x + 4y = 52$ and $20x + 4y = 64$ are contradictory (both cannot be true simultaneously), System 1 has no solution.

System 2

Equations:

1. $5x + y = 13$
2. $20x = 52 - 4y$

Let's manipulate the second equation to see if it aligns with the first:

• Rewrite the second equation in a similar form: $20x + 4y = 52$
• Now, if we divide the second equation by 4, we get: $5x + y = 13$
• This is the same as the first equation in System 2, which means the two equations are essentially the same line.
• Thus, System 2 has infinitely many solutions since both equations represent the same line.

Conclusion

• System 1 has no solution.
• System 2 has many (infinitely) solutions.

Would you like further clarification on any steps or have more questions?

Here are some related questions:

1. How can you tell if two lines are parallel or identical without graphing?
2. What does it mean for a system to have no solution?
3. What other methods can we use to identify the type of solution a system has?
4. How does multiplying an equation affect its solutions?
5. What does it mean for a system to have infinitely many solutions?

Tip: When analyzing systems of equations, look for simple manipulations like scaling or rearranging to see if the equations align or contradict.