Equation of a Straight Line

The gradient $m$ is defined as the ratio of the change in the vertical coordinate to the change in the horizontal coordinate, expressed as $m = \frac{y_2 - y_1}{x_2 - x_1}$. This value represents the slope's magnitude and direction: positive gradients rise from left to right, while negative gradients fall.

The principle of collinearity states that if three or more points lie on the same straight line, the gradient between any two pairs of those points must be identical. This is a powerful tool for verifying if a set of coordinates belongs to a single linear path.

The relationship between two lines is determined by their gradients. Lines are parallel if they share the same gradient ($m_1 = m_2$), and they are perpendicular if the product of their gradients is $-1$ ($m_1 \cdot m_2 = -1$), meaning one is the negative reciprocal of the other.

• Slope-Intercept Form ($y = mx + c$): This is the most common form, where $m$ is the gradient and $c$ is the y-intercept. It is ideal for quickly sketching a line or identifying its behavior at a glance.

• Point-Slope Form ($y - y_1 = m(x - x_1)$): This method is highly efficient when you know the gradient and a single point $(x_1, y_1)$ through which the line passes. It avoids the need to solve for $c$ as an intermediate step.

• General Form ($ax + by + c = 0$): In this form, $a, b,$ and $c$ are typically integers. This is often required for final answers in exams and is useful for finding intercepts by setting $x=0$ or $y=0$ respectively.

• Parallel vs. Perpendicular: Parallel lines never meet because they change at the exact same rate. Perpendicular lines meet at a $90^{\circ}$ angle, and their gradients are mathematically linked such that if one is $\frac{a}{b}$, the other must be $-\frac{b}{a}$.

• Always Sketch: Even a rough drawing can prevent major errors, such as using a positive gradient for a line that clearly slopes downwards. It helps verify if your calculated intercept and slope 'look' correct.

• Rearrange First: When given an equation like $3x + 2y = 5$, never assume the gradient is the coefficient of $x$. You must rearrange it into $y = mx + c$ form (e.g., $y = -\frac{3}{2}x + \frac{5}{2}$) before identifying the gradient.

• Fraction Management: Keep gradients as exact fractions (e.g., $\frac{2}{3}$) rather than decimals (0.66...). This maintains precision, especially when calculating perpendicular gradients or finding points of intersection.

• Check Collinearity: To prove three points are collinear, calculate the gradient between the first two and the gradient between the last two; if they are equal, the points are on the same line.

• Sign Errors in Point-Slope: A frequent mistake is failing to distribute the negative sign correctly in $y - y_1 = m(x - x_1)$, particularly when $x_1$ or $y_1$ are negative (e.g., $x - (-2)$ becomes $x + 2$).

• Reciprocal Confusion: Students often forget the 'negative' part of the negative reciprocal for perpendicular lines. If $m = 5$, the perpendicular gradient is $-\frac{1}{5}$, not just $\frac{1}{5}$.

• Horizontal and Vertical Lines: Remember that horizontal lines have the form $y = k$ (gradient 0) and vertical lines have the form $x = k$ (gradient undefined). These do not follow the standard $y=mx+c$ format in the same way.