Understanding the Slope of a Line
The slope of a straight line measures its steepness and direction. It is commonly denoted by the letter \( m \). For a line expressed in the form \( y = mx + c \), \( m \) is the coefficient of \( x \). This value tells us how much \( y \) changes for a unit increase in \( x \).
For example, in the line \( y = 4x - 1 \), the slope \( m = 4 \). This means for every increase of 1 in \( x \), \( y \) increases by 4 units.
- Slope \( m \) is the coefficient of \( x \) in \( y = mx + c \).
- Positive slope means the line rises from left to right.
- Negative slope means the line falls from left to right.
Properties of Parallel Lines
Two lines are parallel if they never meet, no matter how far they are extended. A key property is that parallel lines have the same slope.
If one line has slope \( m \), any line parallel to it must also have slope \( m \). This fact allows us to find the equation of a parallel line when we know a point it passes through.
- Parallel lines have equal slopes.
- Different \( y \)-intercepts mean the lines are distinct but parallel.
Using the Point-Slope Form to Find a Parallel Line
The point-slope form of a line's equation is useful when you know a point \( (x_1, y_1) \) on the line and its slope \( m \). The formula is:
\[ y - y_1 = m(x - x_1) \] This equation expresses the relationship between \( x \) and \( y \) for the line passing through \( (x_1, y_1) \) with slope \( m \).
To find the equation of a line parallel to \( y = 4x - 1 \) passing through \( (1, -3) \), use the slope \( m = 4 \) and substitute into the formula:
- Identify slope \( m \) from the original line.
- Use the point \( (x_1, y_1) \) given.
- Substitute into \( y - y_1 = m(x - x_1) \).
Worked Example: Equation of a Parallel Line
Given the line \( y = 4x - 1 \), find the equation of the line parallel to it passing through \( (1, -3) \).
Step 1: Identify the slope of the original line: \( m = 4 \).
Step 2: Use the point-slope form with \( (x_1, y_1) = (1, -3) \): \[ y - (-3) = 4(x - 1) \] which simplifies to \[ y + 3 = 4x - 4 \].
- Subtract 3 from both sides: \( y = 4x - 4 - 3 \).
- Simplify: \( y = 4x - 7 \).
- This is the equation of the parallel line in slope-intercept form.
Rearranging to Slope-Intercept Form
The slope-intercept form \( y = mx + c \) is often preferred because it clearly shows the slope and the \( y \)-intercept.
After using the point-slope form, rearrange the equation by expanding brackets and isolating \( y \) on one side.
This makes it easier to graph the line or compare it with others.
- Expand brackets in point-slope form.
- Isolate \( y \) to write \( y = mx + c \).
- Check that the slope matches the original line's slope for parallelism.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify slope | Find slope \( m \) from given line equation | From \( y = 4x - 1 \), slope \( m = 4 \) | Always extract slope correctly to avoid errors |
| Apply point-slope form | Use \( y - y_1 = m(x - x_1) \) with given point | For point (1, -3), \( y + 3 = 4(x - 1) \) | Use exact coordinates and slope for accuracy |
| Rearrange equation | Expand and isolate \( y \) to get \( y = mx + c \) | \( y = 4x - 7 \) after rearrangement | Final form is easier to interpret and check |
| Check parallelism | Confirm new line slope equals original slope | Both slopes are 4, so lines are parallel | Essential to earn full marks on parallel line questions |
Active recall
Quick check
1. What is the slope of the line \( y = -2x + 5 \)?
The slope \( m = -2 \) because it is the coefficient of \( x \).
2. If a line is parallel to \( y = 3x + 1 \), what is its slope?
The slope is also 3, since parallel lines have equal slopes.
3. Find the equation of the line parallel to \( y = 3x + 1 \) passing through \( (2, 4) \).
Using point-slope form: \( y - 4 = 3(x - 2) \), which simplifies to \( y = 3x - 2 \).
4. Why must parallel lines have the same slope?
Because having the same slope means they rise and run at the same rate, so they never intersect.