Equation of a Straight Line

• A straight line represents a set of points where the change in the vertical coordinate ($y$) is directly proportional to the change in the horizontal coordinate ($x$).

• The gradient (m), or slope, quantifies this steepness and is calculated as the ratio of the 'rise' to the 'run' between any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line.

• The y-intercept (c) is the specific point where the line crosses the y-axis, occurring where the x-coordinate is exactly zero.

• The x-intercept is the point where the line crosses the x-axis, occurring where the y-coordinate is exactly zero.

Standard Forms of the Equation

• Slope-Intercept Form: $y = mx + c$. This is the most common form for graphing as it explicitly shows the gradient ($m$) and the y-intercept ($c$).
• Point-Slope Form: $y - y_1 = m(x - x_1)$. This is the most efficient method for finding an equation when you are given a specific point $(x_1, y_1)$ and the gradient $m$.
• General Form: $ax + by + c = 0$. In this form, $a, b,$ and $c$ are typically expressed as integers. This form is useful for general algebraic manipulation and finding intercepts easily.

Step-by-Step Methodology

• Step 1: Identify the Gradient: If not given, calculate $m$ using two points or the relationship to another line (parallel/perpendicular).
• Step 2: Choose a Point: Select any point $(x_1, y_1)$ that lies on the line.
• Step 3: Substitute into Point-Slope: Use $y - y_1 = m(x - x_1)$ to create the initial equation.
• Step 4: Rearrange: Simplify the equation into the required format ($y=mx+c$ or $ax+by+c=0$).

• Parallel vs. Perpendicular: Parallel lines share the exact same gradient ($m_1 = m_2$), whereas perpendicular lines have gradients that are negative reciprocals of each other ($m_1 \cdot m_2 = -1$).

• Horizontal vs. Vertical: A horizontal line has a gradient of $0$ and an equation of the form $y = k$. A vertical line has an undefined gradient and an equation of the form $x = h$.

• Always Sketch: Before performing calculations, draw a quick sketch of the points or the line. This helps verify if your calculated gradient should be positive or negative and if your intercepts look reasonable.

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

• Verify with Substitution: Once you have found the equation, substitute the coordinates of a known point back into the equation. If the left side equals the right side, your equation is correct.

• Watch the Signs: A common error is failing to distribute a negative gradient correctly across the $(x - x_1)$ term in the point-slope formula.