Equation of a Straight Line

• Gradient-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 immediate graphing.

• Point-Gradient Form: $y - y_1 = m(x - x_1)$. This form is used when you know the gradient $m$ and a specific point $(x_1, y_1)$ that the line passes through.

• General Form: $ax + by + c = 0$. In this form, $a, b,$ and $c$ are typically integers. This is often required for final answers in formal geometry problems.

• Calculation: The gradient is calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. This represents the change in the vertical coordinate divided by the change in the horizontal coordinate.

• Parallel Lines: Two lines are parallel if and only if they have the same gradient ($m_1 = m_2$). They maintain a constant distance and never intersect.

• Perpendicular Lines: Two lines are perpendicular if they meet at a right angle ($90^\circ$). Their gradients are negative reciprocals of each other, satisfying the condition $m_1 \times m_2 = -1$.

• Collinearity: Points are collinear if they lie on the same straight line. This can be verified by checking if the gradient between any two pairs of points is identical.

Finding the Equation from Two Points

• Step 1: Calculate the gradient ($m$) using the two given coordinates $(x_1, y_1)$ and $(x_2, y_2)$.
• Step 2: Substitute the gradient and one of the points into the point-gradient formula $y - y_1 = m(x - x_1)$.
• Step 3: Rearrange the equation into the required format, such as $y = mx + c$ or $ax + by + c = 0$.

Working with Line Segments

• Midpoint: The center of a line segment is found by averaging the coordinates: $M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.
• Length: The distance $d$ between two points is derived from Pythagoras' Theorem: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

• Sketching: Always draw a quick sketch of the line. This helps verify if your calculated gradient (positive vs. negative) and intercept match the visual logic of the problem.

• Fractional Gradients: When dealing with perpendicular lines, if $m = \frac{a}{b}$, the perpendicular gradient is $-\frac{b}{a}$. Always flip the fraction and change the sign.

• Integer Constraints: If an exam asks for the form $ax + by + c = 0$, ensure $a, b,$ and $c$ are integers. Multiply the entire equation by the common denominator to clear any fractions.

• Verification: Plug the coordinates of a point back into your final equation. If the left side equals the right side, your equation is correct.

• Real-world Application: Linear equations model situations where one variable changes at a constant rate relative to another, such as cost over time or distance at a constant speed.

• Interpretation of $m$: In a model, the gradient represents the rate of change (e.g., price per unit or speed).

• Interpretation of $c$: The y-intercept represents the initial value or fixed cost (the value of $y$ when $x = 0$).