We are given two equations involving $x$ and $y$:

1. $y = 2.5x + 3$
2. $2.5x - y + 3 = 0$

Step 1: Analyze the equations

• The first equation is in slope-intercept form $y = mx + c$, where $m = 2.5$ (slope) and $c = 3$ (y-intercept).
• The second equation is in general form. Rearrange it to slope-intercept form to better understand its relationship to the first equation:

$2.5x - y + 3 = 0$

$y = 2.5x + 3$

Step 2: Compare the equations

Notice that the second equation simplifies to the exact same form as the first equation:

$y = 2.5x + 3$

Step 3: Conclusion

Both equations describe the same line. This means:

• They are not two different lines.
• Every point on the line $y = 2.5x + 3$ satisfies both equations.

Key Points:

• The system of equations has infinitely many solutions because the two equations are identical.
• Graphically, they represent the same line.

Would you like a detailed graph or further clarification?

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Relative Questions:

1. What does it mean when two equations represent the same line?
2. How can you determine if two linear equations are identical?
3. What is the importance of slope and intercept in analyzing linear equations?
4. How can systems of linear equations have no solution, one solution, or infinitely many solutions?
5. How would this analysis change if the slopes of the two lines were different?

Tip: When comparing two linear equations, always simplify them to the same form (e.g., slope-intercept form) to check for equality or parallelism.