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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!

Mai is given two systems of linear equations to solve:
System 1:
5x + y = 13
20x + 4y = 64

System 2:
5x + y = 13
20x = 52 - 4y

One system has no solution, and the other has infinitely many solutions. Which system has no solution and which has infinitely many solutions? Explain without graphing the equations.

Learn how to determine whether systems of linear equations have no solution or infinitely many solutions. This problem involves comparing two systems, applying algebraic techniques to identify inconsistent and dependent systems. Suitable for Grades 9-11.

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