15x−5y=20
6x−2y=8  how many solutions

To determine how many solutions exist for the given system of linear equations:

\[
15x - 5y = 20
\]
\[
6x - 2y = 8
\]

we will analyze whether the system is consistent and independent, dependent, or inconsistent. 

### Step 1: Simplify the equations

Start by simplifying each equation:
1. \( 15x - 5y = 20 \) can be simplified by dividing all terms by 5:
   \[
   3x - y = 4
   \]

2. \( 6x - 2y = 8 \) can be simplified by dividing all terms by 2:
   \[
   3x - y = 4
   \]

### Step 2: Compare the simplified equations

The simplified equations are identical:
\[
3x - y = 4
\]
This means that both equations represent the same line. Therefore, there are infinitely many solutions because every point on the line satisfies both equations.

### Conclusion

The system has **infinitely many solutions**.

Would you like a more detailed explanation or have any other questions?

### Related Questions
1. What is the definition of a consistent system of equations?
2. How do you determine if two lines are parallel?
3. What happens if two lines have different slopes?
4. How can you classify a system of equations as independent, dependent, or inconsistent?
5. How do you graphically represent a system of linear equations?

### Tip
When two linear equations simplify to the same equation, it indicates that the system is dependent, meaning the lines are coincident and have infinitely many solutions.

Learn how to determine the number of solutions for the system of equations 15x−5y=20 and 6x−2y=8. This algebra problem involves simplifying linear equations, identifying coincident lines, and classifying systems as dependent. Suitable for students in Grades 8-10.

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