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:
