Equation of a Line

Q: Define the slope of a line. What is the slope of a line inclined at 30° to the positive direction of the x-axis?

The slope of a line is the tangent of the angle the line makes with the positive direction of the x-axis. For 30°, m = tan 30° = (1)/(√(3)).

Q: Find the slope of the line through (3, -2) and (-1, 6). Does the line rise or fall from left to right?

m = (6 - (-2))/(-1 - 3) = (8)/(-4) = -2. The slope is negative, so the line falls.

Q: Explain why the slope of a vertical line is not defined, while the slope of a horizontal line is 0.

For two points on a vertical line, the run x2 - x1 is 0, and division by zero is not defined. On a horizontal line, the rise y2 - y1 is 0 for any non-zero run, so the slope is 0.

Q: Find the slope and y-intercept of 4x - 2y + 6 = 0.

2y = 4x + 6, so y = 2x + 3. Slope 2, y-intercept 3.

Q: Show that (-2, -3), (1, 3) and (3, 7) are collinear.

Slope of the first pair = (3 + 3)/(1 + 2) = 2; slope of the second pair = (7 - 3)/(3 - 1) = 2. The slopes are equal and (1, 3) is common, so the points are collinear.

Q: Find the equation of the line through (-1, 3) and (2, 9).

m = (9 - 3)/(2 + 1) = 2. Then y - 3 = 2(x + 1), so y = 2x + 5.

ICSE Class 10 · Mathematics · Chapter 14

What this chapter covers

Inclination and slope of a line
Slope from two points on a line
Collinear points
The equation of a line and the y-intercept
Slope-intercept form y = mx + c
Two-point form (y - y1) = m(x - x1)
Condition for two lines to be parallel
Condition for two lines to be perpendicular
Simple applications
Common errors and how to avoid them

Key definitions

The inclination of a line is the angle θ that the line makes with the positive direction of the x-axis, measured anticlockwise, where 0^° ≤ θ < 180^°.

The slope of a line is the tangent of its inclination, written m = tanθ, where θ is the angle the line makes with the positive direction of the x-axis.

The gradient of a line is another name for its slope, and both words describe how steeply the line rises or falls.

Collinear points are points that lie on one and the same straight line.

The equation of a line is an equation in x and y that is satisfied by the coordinates of every point on the line and by the coordinates of no other point.

The slope-intercept form is the equation y = mx + c of a straight line, in which m is the slope of the line and c is its y-intercept.

The two-point form is the equation y - y₁ = m(x - x₁) of a line, where (x₁, y₁) is a known point on the line and m is its slope, found either directly or from two given points.

Parallel lines are lines in the same plane that never meet, however far they are extended.

Perpendicular lines are lines that meet each other at a right angle.

A perpendicular bisector is the line that passes through the mid-point of a given line segment and is perpendicular to that segment.

A median of a triangle is the line segment that joins a vertex to the mid-point of the opposite side.

An altitude of a triangle is the perpendicular drawn from a vertex to the opposite side, or to the opposite side produced.

Key points

The inclination θ of a line is the angle it makes with the positive direction of the x-axis, with 0^° ≤ θ < 180^°.
The slope or gradient is m = tanθ, the rise for each unit of run; it is positive for a rising line, negative for a falling line, 0 for a horizontal line and not defined for a vertical line.
Through (x₁, y₁) and (x₂, y₂), the slope is m = (y₂ - y₁)/(x₂ - x₁).
Three points are collinear when the slopes of the two segments sharing a common point are equal.
The y-intercept c is the ordinate of the point where the line meets the y-axis, the point on the line where x = 0.
Slope-intercept form: y = mx + c; a line through the origin is y = mx.
Two-point form: y - y₁ = m(x - x₁), with m given or found from two points.
Lines parallel to the axes are y = b and x = a.
For ax + by + k = 0, make y the subject: the slope is -(a)/(b).
Parallel lines: m₁ = m₂. Perpendicular lines: m₁ m₂ = -1, except when one line is vertical.
To solve an application, find a point and a slope, then write the equation; check by substituting a known point.

Questions answered

Define the slope of a line. What is the slope of a line inclined at 30^° to the positive direction of the x-axis?

The slope of a line is the tangent of the angle the line makes with the positive direction of the x-axis. For 30^°, m = tan 30^° = (1)/(√(3)).

Find the inclination of a line whose slope is √(3).

tanθ = √(3), so θ = 60^°.

Find the slope of the line through (3, -2) and (-1, 6). Does the line rise or fall from left to right?

m = (6 - (-2))/(-1 - 3) = (8)/(-4) = -2. The slope is negative, so the line falls.

Explain why the slope of a vertical line is not defined, while the slope of a horizontal line is 0.

For two points on a vertical line, the run x₂ - x₁ is 0, and division by zero is not defined. On a horizontal line, the rise y₂ - y₁ is 0 for any non-zero run, so the slope is 0.

Find the slope and y-intercept of 4x - 2y + 6 = 0.

2y = 4x + 6, so y = 2x + 3. Slope 2, y-intercept 3.

Write the equation of the line with slope -3 that cuts the y-axis at (0, 5).

y = -3x + 5, or 3x + y = 5.

Show that (-2, -3), (1, 3) and (3, 7) are collinear.

Slope of the first pair = (3 + 3)/(1 + 2) = 2; slope of the second pair = (7 - 3)/(3 - 1) = 2. The slopes are equal and (1, 3) is common, so the points are collinear.

Find the equation of the line through (-1, 3) and (2, 9).

m = (9 - 3)/(2 + 1) = 2. Then y - 3 = 2(x + 1), so y = 2x + 5.

Past board exam questions

ICSE 2027 sample paper · 1 mark

The line y = 2x passes through which pair of coordinates?

(0, 0) and (2, 2)
(0, 2) and (2, 4)
(1, 2) and (-1, -2)
(2, 2) and (-2, -2)

Show answer

(1, 2) and (-1, -2)

ICSE 2026 board paper · 1 mark

Find the equation of a line whose y-intercept is 6 and is parallel to x-axis.

y = 6
x = 6
x + y = 6
y - x = 6

Show answer

y = 6

ICSE 2026 sample paper · 1 mark

Which of the following equations represents a line equally inclined to the axes?

2x – 3y + 7 = 0
x – y = 7
x = 7
y = –7

Show answer

x – y = 7

ICSE 2025 board paper · 1 mark

The equation of the line passing through origin and parallel to the line 3x + 4y + 7 = 0 is:

3x + 4y + 5 = 0
4x - 3y - 5 = 0
4x - 3y = 0
3x + 4y = 0

Show answer

3x + 4y = 0

ICSE 2025 sample paper · 4 marks

A line segment joining P (2, –3) and Q (0, –1) is cut by the x-axis at the point R. A line AB cuts the y axis at T(0,6) and is perpendicular to PQ at S. Find the: (a) equation of line PQ (b) equation of line AB (c) coordinates of points R and S.

Show answer

(a) Slope of PQ = (-1 - (-3))/(0 - 2) = (2)/(-2) = -1. Through Q(0, −1): y + 1 = -1(x - 0), i.e. x + y + 1 = 0. (b) AB ⊥ PQ, so slope of AB = 1. Through T(0, 6): y = x + 6, i.e. x - y + 6 = 0. (c) R is on the x-axis: put y = 0 in x + y + 1 = 0: x = -1. R(−1, 0). S is the intersection of PQ and AB: from y = x + 6 and x + y = -1: 2x + 6 = -1, x = -(7)/(2), y = (5)/(2). S(-(7)/(2), (5)/(2)).

ICSE 2025 board paper · 3 marks

P and Q are points such that PQ cuts off intercepts of 5 units and 3 units along the x-axis and y-axis respectively, with P on the positive x-axis and Q on the negative y-axis. Line RS is perpendicular to PQ and passes through the origin. Find the: (a) coordinates of P and Q (b) equation of line RS

Show answer

(a) P lies on the positive x-axis and Q on the negative y-axis: P(5, 0), Q(0, –3). (b) Slope of PQ = (-3 - 0)/(0 - 5) = (3)/(5). RS ⊥ PQ, so slope of RS = -(5)/(3). RS passes through (0, 0): y = -(5)/(3)x ⇒ 5x + 3y = 0.

ICSE 2025 compartment paper · 3 marks

Equation of a line AB is x + 2y + 6 = 0. A perpendicular PQ is dropped on AB from the point P(3, –2) meeting AB at Q. Find the: (a) equation of PQ. (b) coordinates of the point Q.

Show answer

(a) Slope of AB = -(1)/(2), so slope of PQ = 2. PQ: y + 2 = 2(x - 3) ⇒ y = 2x - 8, i.e. 2x - y - 8 = 0. (b) Solving with AB: x + 2(2x - 8) + 6 = 0 ⇒ 5x - 10 = 0 ⇒ x = 2, y = 2(2) - 8 = -4. Q(2, –4).

ICSE 2024 board paper · 1 mark

Points A (x, y), B (3, -2) and C (4, -5) are collinear. The value of y in terms of x is:

3x - 11
11 - 3x
3x - 7
7 - 3x

Show answer

7 - 3x