To determine whether the lines represented by the equations [tex]\( 6x - 2y = -2 \)[/tex] and [tex]\( y = 3x + 12 \)[/tex] are perpendicular, parallel, or neither, follow these steps:
1. Convert the first equation to slope-intercept form [tex]\( y = mx + b \)[/tex].
- Start with the equation: [tex]\( 6x - 2y = -2 \)[/tex].
- Solve for [tex]\( y \)[/tex]:
[tex]\[
-2y = -6x - 2
\][/tex]
Divide every term by -2:
[tex]\[
y = 3x + 1
\][/tex]
- The slope ([tex]\( m \)[/tex]) of the first line is 3.
2. The slope of the second line [tex]\( y = 3x + 12 \)[/tex] is already given as the coefficient of [tex]\( x \)[/tex].
- Thus, the slope ([tex]\( m \)[/tex]) of the second line is also 3.
3. Compare the slopes to determine the relationship between the lines:
- If the slopes are equal, the lines are parallel.
- If the product of the slopes is -1, the lines are perpendicular.
- If neither condition is satisfied, the lines are neither parallel nor perpendicular.
In this case, both lines have a slope of 3. Since the slopes are the same, the lines are parallel.
Therefore:
The comparison of their slopes is equal, so the lines are parallel.
