Question
y=3x+6
y=-13x-4
one solution, no solution, or infinite solutions
y=-13x-4
one solution, no solution, or infinite solutions
Answers
GPT-4o mini
To determine whether the two equations \( y = 3x + 6 \) and \( y = -13x - 4 \) have one solution, no solutions, or infinite solutions, we can analyze their slopes and intercepts.
1. **Identify the slopes:**
- The slope of the first equation \( y = 3x + 6 \) is \( 3 \).
- The slope of the second equation \( y = -13x - 4 \) is \( -13 \).
2. **Compare the slopes:**
- The slopes are different (\( 3 \neq -13 \)). This indicates that the lines are not parallel.
Since the lines are not parallel, they will intersect at exactly one point, which means there is one unique solution for the system of equations.
**Conclusion:** The system of equations has **one solution**.
1. **Identify the slopes:**
- The slope of the first equation \( y = 3x + 6 \) is \( 3 \).
- The slope of the second equation \( y = -13x - 4 \) is \( -13 \).
2. **Compare the slopes:**
- The slopes are different (\( 3 \neq -13 \)). This indicates that the lines are not parallel.
Since the lines are not parallel, they will intersect at exactly one point, which means there is one unique solution for the system of equations.
**Conclusion:** The system of equations has **one solution**.