To determine the relationship between the two lines, we need to find their slopes.
Consider the following lines given by their equations:
1. [tex]\( 6x - 2y = -2 \)[/tex]
2. [tex]\( y = 3x + 12 \)[/tex]
Step 1: Convert the first equation to slope-intercept form [tex]\( y = mx + b \)[/tex]
The first equation is:
[tex]\[ 6x - 2y = -2 \][/tex]
To convert this to slope-intercept form:
1. Isolate [tex]\( y \)[/tex] on one side:
[tex]\[ -2y = -6x - 2 \][/tex]
2. Divide every term by [tex]\(-2\)[/tex]:
[tex]\[ y = 3x + 1 \][/tex]
Step 2: Determine the slopes of the equations
Now that both equations are in slope-intercept form, we can identify their slopes:
- For the equation [tex]\( y = 3x + 1 \)[/tex], the slope ([tex]\( m_1 \)[/tex]) is 3.
- For the equation [tex]\( y = 3x + 12 \)[/tex], the slope ([tex]\( m_2 \)[/tex]) is 3.
Step 3: Compare the slopes to determine the relationship
- Two lines are parallel if their slopes are equal ([tex]\( m_1 = m_2 \)[/tex]).
- Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex] ([tex]\( m_1 \cdot m_2 = -1 \)[/tex]).
- If neither condition is satisfied, the lines are neither parallel nor perpendicular.
Since the slopes we found are:
[tex]\[ m_1 = 3, \, m_2 = 3 \][/tex]
The slopes are equal, meaning the lines are parallel.
Therefore, the lines are parallel.
