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**.