Equation of a line parallel to a given line
Equation of a Line Parallel to a Given Line
Understanding the equation of a line parallel to a given line is a fundamental concept in coordinate geometry. When two lines are parallel, they have the same slope but different y-intercepts. This means that the lines never intersect and are always the same distance apart.
Slope-Intercept Form
The slope-intercept form of a line is given by:
$$ y = mx + b $$
where:
- $m$ is the slope of the line
- $b$ is the y-intercept, the point where the line crosses the y-axis
Condition for Parallel Lines
For two lines to be parallel, their slopes must be equal. If we have a line with the equation $y = mx + b$, any line parallel to it will have the form:
$$ y = mx + c $$
where $c$ is a different y-intercept.
Table of Differences and Important Points
| Aspect | Given Line | Parallel Line |
|---|---|---|
| Equation | $y = mx + b$ | $y = mx + c$ |
| Slope | $m$ | $m$ (same as given) |
| Y-intercept | $b$ | $c$ (different) |
| Intersection | Crosses y-axis at $b$ | Crosses y-axis at $c$ |
| Distance to Origin | Varies | Varies |
| Angle with X-axis | Constant | Constant (same angle) |
Formulas
- Slope of the given line: $m$
- Slope of the parallel line: $m$ (same as given line)
- Equation of the parallel line: $y = mx + c$
Examples
Example 1: Finding a Parallel Line
Given the line $y = 2x + 3$, find the equation of a line parallel to it that passes through the point $(4, 1)$.
Solution:
The slope of the given line is $m = 2$. A line parallel to it will also have a slope of $m = 2$.
Using the point-slope form of the line equation, which is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line, we can substitute the point $(4, 1)$ and the slope $m = 2$:
$$ y - 1 = 2(x - 4) $$
Expanding and simplifying:
$$ y - 1 = 2x - 8 $$ $$ y = 2x - 7 $$
So, the equation of the line parallel to $y = 2x + 3$ and passing through the point $(4, 1)$ is $y = 2x - 7$.
Example 2: Verifying Parallel Lines
Verify if the line $y = -3x + 5$ is parallel to the line $y = -3x - 4$.
Solution:
Both lines have the slope $m = -3$. Since the slopes are equal, the lines are parallel. The y-intercepts are different ($5$ and $-4$), which confirms that the lines will never intersect.
Practice Problems
- Find the equation of a line parallel to the line $4x - 2y = 7$ that passes through the point $(0, -3)$.
- Determine if the lines $y = \frac{1}{2}x + 6$ and $y = \frac{1}{2}x - 3$ are parallel.
- Write the equation of a line parallel to $y = -\frac{3}{4}x + 2$ with a y-intercept of $-5$.
By understanding the concept of parallel lines and their equations, you can solve various problems related to coordinate geometry and straight lines. Remember that the key characteristic of parallel lines is that they have identical slopes.