Question

to determine which pair of equations form a system
with one solution, no solution, or infinite solutions.
y=3x+1

Answer 1

To determine the characteristics of a system of equations, you typically need two equations to analyze their relationship. The nature of the solutions can be classified as follows:

1. **One solution**: The lines represented by the equations intersect at exactly one point. This occurs when the equations have different slopes.
2. **No solution**: The equations represent parallel lines that never intersect. This happens when the equations have the same slope but different y-intercepts.
3. **Infinite solutions**: The equations represent the same line. This occurs when one equation is a multiple of the other.

You provided one equation:

\[ y = 3x + 1 \]

To identify the type of system this equation may form, we need a second equation. Here are examples of pairs that illustrate each scenario:

1. **One solution**:
- Example: \( y = -2x + 4 \) (different slope than \( 3 \))
- System:
- \( y = 3x + 1 \)
- \( y = -2x + 4 \)

2. **No solution**:
- Example: \( y = 3x - 5 \) (same slope as \( 3 \) but different y-intercept)
- System:
- \( y = 3x + 1 \)
- \( y = 3x - 5 \)

3. **Infinite solutions**:
- Example: \( y = 3x + 1 \) (the same line)
- System:
- \( y = 3x + 1 \)
- \( y = 6x + 2 \) (which simplifies to the same equation)

If you have a specific second equation in mind, please provide it, and I can help you analyze the system further!