Slope From the Equation of a Straight Line

1. Concept Overview

The slope (gradient) of a straight line represents its inclination, i.e., how steep the line is and whether it is increasing or decreasing.

When the equation of a line is given, the slope can be identified or extracted depending on the form of the equation.

2. Slope from Different Forms of Equation

(A) Slope–Intercept Form

Equation:
[y = mx + c]

• Slope [m] is directly visible
• No calculation required

Slope = coefficient of x

(B) Point–Slope Form

Equation:
[y − y_1 = m(x − x_1)]

• Slope is explicitly written

Slope = m

(C) Two-Point Form

Equation:
[\dfrac{y − y_1}{y_2 − y_1} = \dfrac{x − x_1}{x_2 − x_1}]

Slope:
[m = \dfrac{y_2 − y_1}{x_2 − x_1}]

(D) General Form

Equation:
[Ax + By + C = 0]

Rewriting:
[By = −Ax − C]
[y = −\dfrac{A}{B}x − \dfrac{C}{B}]

Slope:
[m = −\dfrac{A}{B}], provided [B ≠ 0]

(E) Intercept Form

Equation:
[\dfrac{x}{a} + \dfrac{y}{b} = 1]

Rewriting:
[y = −\dfrac{b}{a}x + b]

Slope:
[m = −\dfrac{b}{a}]

3. Special Cases (Very Important for Exams)

4. Key Conceptual Observations

• Positive slope ⇒ line rises from left to right
• Negative slope ⇒ line falls from left to right
• Zero slope ⇒ horizontal line
• Undefined slope ⇒ vertical line

Slope depends on:

• Ratio of change in y to change in x
• Not on where the line is placed

5. Worked Examples

Example 1

Find the slope of the line:
[3x − 2y + 7 = 0]

Solution:

1. Compare with [Ax + By + C = 0]
2. Here [A = 3], [B = −2]
3. Slope:
   [m = −\dfrac{A}{B} = −\dfrac{3}{−2}]

Answer:
[m = \dfrac{3}{2}]

Example 2

Find the slope of:
[\dfrac{x}{4} + \dfrac{y}{−2} = 1]

Solution:

1. Intercept form ⇒ [a = 4], [b = −2]
2. Slope:
   [m = −\dfrac{b}{a} = −\dfrac{−2}{4}]

Answer:
[m = \dfrac{1}{2}]

Example 3

Find the slope of the line:
[x = 5]

Solution:

• Line is parallel to y-axis

Answer:
Slope is not defined

6. Common Exam Traps (Read Carefully)

• Forgetting the negative sign in [m = −A/B]
• Assuming slope exists for [x = constant]
• Confusing slope with y-intercept
• Cancelling coefficients before identifying A and B

7. Summary Table