1. Understanding Parallel Lines
Parallel lines are lines that never intersect, no matter how far they are extended. They stay the same distance apart everywhere and have the same direction.
On a Cartesian plane, the concept of slope helps us identify whether two lines are parallel.
1.1. Definition
Two lines are said to be parallel if they have the same slope and different y-intercepts.
2. Condition for Parallel Lines
The slopes of the lines decide whether they are parallel. If two lines have slopes \(m_1\) and \(m_2\), then:
2.1. Slope Condition
\( m_1 = m_2 \)
This means both lines rise and fall at the same rate.
2.2. Explanation
If the slopes are equal, the angle of inclination of both lines is the same. This makes the lines move in the same direction, so they never meet.
2.2.1. Important Note
If the slopes are equal but the y-intercepts are also equal, then the two lines are not just parallel—they are the same line.
3. Parallel Lines in Different Forms of Equations
It is easy to check if two lines are parallel once we write their equations in slope-intercept form \(y = mx + c\).
3.1. Slope-Intercept Form
Two lines:
\( y = m_1x + c_1 \)
\( y = m_2x + c_2 \)
are parallel if \(m_1 = m_2\).
3.2. General Form
For lines in the form \(Ax + By + C = 0\), slopes are:
\( m = -\dfrac{A}{B} \)
Thus, two lines are parallel if:
\( -\dfrac{A_1}{B_1} = -\dfrac{A_2}{B_2} \)
4. Examples
These examples help understand how to check if two lines are parallel.
4.1. Example 1: Using Slope-Intercept Form
Lines:
\( y = 3x + 2 \)
\( y = 3x - 5 \)
Both lines have slope 3 → they are parallel.
4.2. Example 2: Using General Form
Lines:
\( 2x + 3y - 4 = 0 \)
\( 4x + 6y + 1 = 0 \)
Slopes:
\( m_1 = -\dfrac{2}{3},\quad m_2 = -\dfrac{4}{6} = -\dfrac{2}{3} \)
Since slopes are equal, these lines are parallel.
4.3. Example 3: Not Parallel
Lines:
\( y = 2x + 7 \)
\( y = -x + 4 \)
Slopes 2 and -1 are not equal → lines are not parallel.