Question
Determine the number of solutions that the following system of linear equations have: Click Here for Help Video
y
=
6
x
+
2
y=6x+2
y
=
−
6
x
+
2
y=−6x+2
Answers
GPT-4o mini
To determine the number of solutions to the given system of linear equations, we can analyze the two equations:
1. \( y = 6x + 2 \)
2. \( y = -6x + 2 \)
Both equations are in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
**For the first equation**:
- Slope \( m_1 = 6 \)
- Y-intercept \( b_1 = 2 \)
**For the second equation**:
- Slope \( m_2 = -6 \)
- Y-intercept \( b_2 = 2 \)
### Step 1: Compare the slopes
The slopes of the two lines are different:
- \( m_1 = 6 \) (first equation)
- \( m_2 = -6 \) (second equation)
### Step 2: Determine the y-intercepts
Both equations have the same y-intercept:
- \( b_1 = b_2 = 2 \)
### Conclusion
Since the two lines have different slopes, they are not parallel and will intersect at exactly one point. Therefore, the system of equations has exactly **one solution**.
In summary, the system of equations has **one solution**.
1. \( y = 6x + 2 \)
2. \( y = -6x + 2 \)
Both equations are in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
**For the first equation**:
- Slope \( m_1 = 6 \)
- Y-intercept \( b_1 = 2 \)
**For the second equation**:
- Slope \( m_2 = -6 \)
- Y-intercept \( b_2 = 2 \)
### Step 1: Compare the slopes
The slopes of the two lines are different:
- \( m_1 = 6 \) (first equation)
- \( m_2 = -6 \) (second equation)
### Step 2: Determine the y-intercepts
Both equations have the same y-intercept:
- \( b_1 = b_2 = 2 \)
### Conclusion
Since the two lines have different slopes, they are not parallel and will intersect at exactly one point. Therefore, the system of equations has exactly **one solution**.
In summary, the system of equations has **one solution**.