Equations of Straight Lines
Different Forms of Linear Equations
Linear equations can be expressed in various forms, showing different properties of the line, such as intercepts, gradients, and points:
- Gradient-Intercept Form: $y = mx + c$
- $m$ represents the gradient (slope) of the line
- $c$ is the y-intercept (where the line crosses the y-axis)
- General Form: $ax + by + d = 0$
- $a$, $b$, and $d$ are constants
- Can be rearranged to gradient-intercept form: $y = -\frac{a}{b}x - \frac{d}{b}$
- Point-Gradient Form: $y - y_1 = m(x - x_1)$
- $(x_1, y_1)$ is a point on the line
- $m$ is the gradient
Let's consider a line passing through the point (2, 5) with a gradient of 3:
- Point-Gradient form: $y - 5 = 3(x - 2)$
- Expanding: $y = 3x - 6 + 5$
- Gradient-Intercept form: $y = 3x - 1$
- General form: $3x - y - 1 = 0$
Gradient and Intercepts
The gradient (m) represents the steepness of a line and is calculated as:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.
NoteThe gradient is positive for lines sloping upwards from left to right, and negative for lines sloping downwards from left to right.
Intercepts are the points where a line crosses the axes:
- y-intercept: where the line crosses the y-axis (x = 0)
- x-intercept: where the line crosses the x-axis (y = 0)
For the line $y = 2x + 3$:
- Gradient: $m = 2$
- y-intercept: $(0, 3)$
- x-intercept: $(-\frac{3}{2}, 0)$ (found by setting $y = 0$ and solving for x)
This gradient formula can be substituted into the gradient-intercept and point-gradient forms of the line, yielding:
$$y = \frac{y_2-y_1}{x_2-x_1}x+c$$
and
$$y - y_1 = \frac{y_2-y_1}{x_2-x_1}(x - x_1)$$
Parallel and Perpendicular Lines
Parallel Lines
Parallel lines never intersect and have the same gradient. If two lines with equations $y = m_1x + c_1$ and $y = m_2x + c_2$ are parallel, then:
$$ m_1 = m_2 $$
ExampleThe lines $y = 3x + 2$ and $y = 3x - 5$ are parallel because they have the same gradient (3).
Common MistakeStudents often forget that parallel lines can have different y-intercepts. The lines $y = 2x + 1$ and $y = 2x + 4$ are parallel despite having different y-intercepts.
Perpendicular Lines
Perpendicular lines intersect at right angles (90°). If two lines with gradients $m_1$ and $m_2$ are perpendicular, then:
$$ m_1 \cdot m_2 = -1 $$
This means that the gradients of perpendicular lines are negative reciprocals of each other.
NoteTo understand this topic, grab a pen and a piece of paper.
Draw a straight line and label the change in $y$-coordinate as $a$ $x$-coordinate as $b$, with arrows to show the direction of rise and run. Here, the gradient of the line is $\frac{a}{b}$.
Rotate it $90$ degrees anticlockwise: The new change in $y$ coordinate should be $b$, and the change in $x$ coordinate should be $-a$, so the new gradient is $-\frac{b}{a}$.
So the gradient of the perpendicular line is $-\frac{b}{a}$.
ExampleIf one line has a gradient of 2, a perpendicular line would have a gradient of $-\frac{1}{2}$.
Line 1: $y = 2x + 3$ Line 2: $y = -\frac{1}{2}x + 1$
These lines are perpendicular.
TipTo quickly find the equation of a perpendicular line passing through a point:
- Find the negative reciprocal of the original line's gradient
- Use the point-gradient form with the new gradient and given point
