We are given two systems of linear equations, and Mai claims that one system has no solution, while the other has infinitely many solutions.

System 1:

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

System 2:

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

Let's solve them step by step to determine which system has no solution and which has infinitely many.

Analyzing System 1:

The two equations are:

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

We can multiply the first equation by 4 to make the coefficients of $y$ in both equations comparable:

$4(5x + y) = 4(13)$ $20x + 4y = 52$

Now, we compare this with the second equation of System 1: $20x + 4y = 64$

These two equations are:

1. $20x + 4y = 52$
2. $20x + 4y = 64$

Since these two lines are parallel (same left-hand side, different right-hand side), there is no solution for System 1. The system is inconsistent.

Analyzing System 2:

The equations in System 2 are:

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

First, simplify the second equation: $20x + 4y = 52$

Now, multiply the first equation by 4 (just like we did in System 1): $4(5x + y) = 4(13) \Rightarrow 20x + 4y = 52$

Both equations are identical: $20x + 4y = 52$ $20x + 4y = 52$

This means System 2 represents the same equation twice, so it has infinitely many solutions (the equations are dependent).

Conclusion:

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

Would you like more details or further explanation?

Here are 5 related questions you can explore:

1. How do you identify inconsistent systems of linear equations?
2. What does it mean for a system of equations to have infinitely many solutions?
3. How would you solve these systems algebraically using substitution or elimination?
4. Can you think of a method to classify systems of equations based on their solutions?
5. What are some real-world scenarios where you might encounter a system with no solution or infinitely many solutions?

Tip: For systems of linear equations, parallel lines indicate no solution, while identical lines indicate infinitely many solutions!