9.1 Gradient of a Straight Line
Definition and Formula
The gradient ($m$) of a straight line is the ratio of the vertical distance to the horizontal distance between any two points on the line.
- Given two points $P(x_1, y_1)$ and $Q(x_2, y_2)$:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
- Given the $x$-intercept and $y$-intercept of a line:
$$m = -\frac{y\text{-intercept}}{x\text{-intercept}}$$
9.2 Equation of a Straight Line
Gradient Form ($y = mx + c$)
The general equation of a non-vertical straight line is written as:
$$y = mx + c$$
- $m$ = gradient of the straight line
- $c$ = $y$-intercept (the value of $y$ where the line crosses the $y$-axis, i.e., $x = 0$)
Special Cases
- Horizontal Line (parallel to $x$-axis): Gradient $m = 0$, Equation: $y = c$
- Vertical Line (parallel to $y$-axis): Gradient $m$ is undefined, Equation: $x = a$
Finding the Equation of a Straight Line
- Given gradient $m$ and a point $(x_1, y_1)$:
Substitute $m$, $x_1$, and $y_1$ into $y = mx + c$ to solve for $c$, then write $y = mx + c$.
- Given two points $(x_1, y_1)$ and $(x_2, y_2)$:
First find $m = \frac{y_2 - y_1}{x_2 - x_1}$, then substitute $m$ and one point into $y = mx + c$ to solve for $c$.
9.3 Parallel Lines
Condition for Parallelism
Two straight lines $L_1$ ($y = m_1 x + c_1$) and $L_2$ ($y = m_2 x + c_2$) are parallel if and only if their gradients are equal:
$$m_1 = m_2$$
9.4 Intersection Point of Two Straight Lines
The point of intersection between two non-parallel straight lines can be found by solving their linear equations simultaneously using either:
- Substitution Method
- Elimination Method