Revision: Algebra >> Equation of a Line Maths (English Medium) ICSE Class 10 CISCE

An equation of the form ax + by + c = 0 represents a straight line and is known as a linear equation.

The angle of inclination, or simply the inclination of a line, is the angle θ that the part of the line above the x-axis makes with the positive direction of the x-axis and is measured in an anticlockwise direction.

• Anticlockwise → Positive inclination

• Clockwise → Negative inclination

The slope m of a line is m = tan⁡θ

where θ is the inclination of the line with the positive x-axis.

x-intercept: Point where a line cuts the x-axis, y = 0

y-intercept: Point where a line cuts the y-axis, x = 0

\[m=\frac{y_2-y_1}{x_2-x_1}\]

For line ax + by + c = 0

x-intercept:

\[\left(-\frac{c}{a},0\right)\]

y-intercept:

\[\left(0,-\frac{c}{b}\right)\]

When slope and y-intercept are given

y = mx + c​

• m = slope

• c = y-intercept (value of y when x = 0)

When two points are given

\[\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}\]​​

When the slope and one point are given

y − y₁ = m(x − x₁)

One line has a slope m = tan⁡θ​

The other equally inclined line has a slope m = − tanθ​

Slopes are equal in magnitude, opposite in sign

Standard Results

• x-axis / parallel to x-axis → θ = 0^∘

• y-axis / parallel to y-axis → θ=90^∘

Line Types

• Horizontal line → parallel to x-axis

• Vertical line → parallel to y-axis

• Oblique line → neither parallel to the x-axis nor the y-axis

Nature of Slope

• m > 0 → rising line

• m < 0 → falling line

• m = 0 → horizontal line

• m = ∞→ vertical line

Parallel Lines 

Two lines are parallel ⇔ , their slopes are equal, m₁ = m₂

​Perpendicular Lines

Two lines are perpendicular ⇔ m₁ × m₂ = −1

​Collinearity of Three Points

Points A, B, and C are collinear

Method 1: Distance method

AB + BC = AC

Method 2: Slope method

Slope of AB = Slope of BC

• x-intercept:
  Right of origin → positive
  Left of origin → negative

• y-intercept:
  Above origin → positive
  Below origin → negative