General Form Ax + By + C = 0: Slope and Intercepts

Grades 9-10

Question: explain General Form Ax + By + C = 0 (A, B, C ∈ ℤ and A > 0) Fast Coefficients Read (Paper 2): • Slope m = −A/B • y-intercept = −C/B • x-intercept = −C/A

In the HKDSE Compulsory Part, the General Form of a straight line is the most standard way to express a linear relation. While the Slope-Intercept form (

y = mx + c

) is intuitive for graphing, the General Form

Ax + By + C = 0

is often used in coordinate geometry questions involving circles and intersections.

1. Derivation and Algebraic Transformation

To understand why the "fast coefficients" work, we transform the General Form into the Slope-Intercept form.

\begin{align*} Ax + By + C &= 0 \\ By &= -Ax - C \\ y &= \left( -\frac{A}{B} \right)x + \left( -\frac{C}{B} \right) \end{align*}

By comparing this to the standard

y = mx + c

Slope ( $m$ ): The coefficient of $x$ , which is $-\frac{A}{B}$ .
$y$ -intercept ( $c$ ): The constant term, which is $-\frac{C}{B}$ .

2. Calculating Intercepts

An intercept occurs when the line crosses one of the axes.

$x$ -intercept: Occurs when $y = 0$ . $\begin{align*} Ax + B(0) + C &= 0 \\ Ax &= -C \\ x &= -\frac{C}{A} \end{align*}$
$y$ -intercept: Occurs when $x = 0$ . $\begin{align*} A(0) + By + C &= 0 \\ By &= -C \\ y &= -\frac{C}{B} \end{align*}$

3. Examiner's Tips for Paper 2 (Multiple Choice)

In the DSE Paper 2, speed is critical. You should be able to identify the properties of a line at a glance:

Sign of Slope: If $A$ and $B$ have the same sign, the slope is negative (the line goes "downhill"). If they have opposite signs, the slope is positive ("uphill").
Vertical/Horizontal Lines:
- If $A = 0$ , the line is horizontal ( $y = -\frac{C}{B}$ ).
- If $B = 0$ , the line is vertical ( $x = -\frac{C}{A}$ ).
The "A > 0" Convention: By keeping $A$ positive, the sign of $B$ directly tells you the direction of the line. For example, in $3x - 4y + 5 = 0$ , $A=3$ (positive) and $B=-4$ (negative), so the slope is $-(\frac{3}{-4}) = \frac{3}{4}$ , which is positive.

Visual Guide

In the diagram below, you can manipulate the values of

A, B,

and

C

The Blue Line represents the equation $Ax + By + C = 0$ .
The Red Point indicates the $y$ -intercept $(0, -\frac{C}{B})$ .
The Green Point indicates the $x$ -intercept $(-\frac{C}{A}, 0)$ .
Observe how changing $B$ affects the steepness (slope) and how changing $C$ shifts the line without changing its slope.