What does x = k look like on the Cartesian plane (Class 9)?

When x = k is treated as a linear equation in two variables, it represents a vertical line parallel to the y-axis. This line passes through the point (k, 0) on the x-axis and includes all points of the form (k, y) for every real number y. For example, x = 3 is a vertical line passing through (3, 0).

What is the equation of a line parallel to the x-axis passing through (a, b)?

The equation of a line parallel to the x-axis passing through any point (a, b) is y = b. The y-coordinate is constant along such a line, while x can be any value. For example, the line parallel to the x-axis through (3, −4) has the equation y = −4.

How do you solve Question 3 of Exercise 6.5 — 3x + 2 = 8x − 8?

Solving 3x + 2 = 8x − 8: rearrange to get 2 + 8 = 8x − 3x, which gives 10 = 5x, so x = 2. On the number line, mark the point 2. On the Cartesian plane, this is the vertical line x = 2, which is parallel to the y-axis and passes through (2, 0).

What is the difference between representing x = 3 as one variable vs two variables?

As a one-variable equation, x = 3 has a unique solution: a single point at position 3 on the number line. As a two-variable equation (implicitly x + 0·y = 3), it has infinitely many solutions — all points (3, y) for any y value — which form a vertical straight line on the Cartesian plane, parallel to the y-axis.

Exercise 6.5 — Lines Parallel to Axes | Class 9 Maths | Linear Equations in Two Variables

Lines Parallel to the X-Axis and Y-Axis

In Chapter 6 of Class 9 Mathematics, students explore how linear equations in two variables can be graphed on the Cartesian plane. Exercise 6.5 focuses on a special family of equations — those of the form x = k and y = k — which produce lines that are perfectly parallel to the y-axis or parallel to the x-axis respectively.

Every such equation can be viewed in two ways: as a linear equation in one variable (giving a single point on the number line) or as a linear equation in two variables (giving an infinite vertical or horizontal line on the Cartesian plane). Understanding this duality is the central skill tested in Exercise 6.5.

Key Rules — Equations of Lines Parallel to Axes

Equation Form	Graph Type	Passes Through	Example
x = k	Vertical line — parallel to y-axis	Point (k, 0) on x-axis	x = 3 → vertical line at x = 3
y = k	Horizontal line — parallel to x-axis	Point (0, k) on y-axis	y = −3 → horizontal line at y = −3
x = 0	The y-axis itself	Origin (0, 0)	Equation of the y-axis
y = 0	The x-axis itself	Origin (0, 0)	Equation of the x-axis

x = a | y = b Line parallel to y-axis passing through (a, 0) · Line parallel to x-axis passing through (0, b)

Coordinate Diagram — x = 3 and y = −3

Blue vertical line: x = 3 (parallel to y-axis) · Orange horizontal line: y = −3 (parallel to x-axis)

Dual Interpretation: The equation x = 3 as a one-variable equation has the unique solution x = 3 (a single point on the number line). As a two-variable equation, it has infinitely many solutions — all points of the form (3, y) for any value of y — forming a vertical line.

Exercise 6.5 — All Problems with Solutions

Question 1

Give the graphical representation of each equation (a) on the number line and (b) on the Cartesian plane.

(i) x = 3

(a) On number line: Mark the point x = 3

(b) On Cartesian plane: Vertical line parallel to y-axis passing through (3, 0)

Equation: x = 3 → vertical line at x = 3

(ii) y + 3 = 0

y + 3 = 0 ⟹ y = −3

(a) On number line: Mark the point y = −3

(b) On Cartesian plane: Horizontal line parallel to x-axis passing through (0, −3)

Equation: y = −3 → horizontal line at y = −3

(iii) y − 4 = 0

y − 4 = 0 ⟹ y = 4

(a) On number line: Mark the point y = 4

(b) On Cartesian plane: Horizontal line parallel to x-axis passing through (0, 4)

Equation: y = 4 → horizontal line at y = 4

(iv) 2x − 9 = 0

2x − 9 = 0 ⟹ 2x = 9 ⟹ x = 9/2

(a) On number line: Mark the point x = 9/2 = 4.5

(b) On Cartesian plane: Vertical line parallel to y-axis passing through (9/2, 0)

Equation: x = 9/2 → vertical line at x = 4.5

(v) 3x + 5 = 0

3x + 5 = 0 ⟹ 3x = −5 ⟹ x = −5/3

(a) On number line: Mark the point x = −5/3 ≈ −1.67

(b) On Cartesian plane: Vertical line parallel to y-axis passing through (−5/3, 0)

Equation: x = −5/3 → vertical line at x = −5/3

Question 2

Give the graphical representation of 2x − 11 = 0 as an equation in (i) one variable and (ii) two variables.

2x − 11 = 0 ⟹ 2x = 11 ⟹ x = 11/2

(i) One variable: x = 11/2 — a single point at 5.5 on the number line

(ii) Two variables: Vertical line parallel to y-axis

Line passes through (11/2, 0) = (5.5, 0) on the Cartesian plane

Question 3

Solve 3x + 2 = 8x − 8 and represent the solution on (i) the number line and (ii) the Cartesian plane.

3x + 2 = 8x − 8

2 + 8 = 8x − 3x

10 = 5x ⟹ x = 2

(i) Number line: Mark the point 2

(ii) Cartesian plane: Vertical line parallel to y-axis

Line x = 2 passes through (2, 0) on the Cartesian plane

Question 4

Write the equation of the line parallel to the x-axis and passing through each point.

Rule: Line parallel to x-axis through (a, b) has equation y = b

(i) (0, −3) → y = −3

(ii) (0, 4) → y = 4

(iii) (2, −5) → y = −5

(iv) (3, −4) → y = −4

Question 5

Write the equation of the line parallel to the y-axis and passing through each point.

Rule: Line parallel to y-axis through (a, b) has equation x = a

(i) (−4, 0) → x = −4

(ii) (2, 0) → x = 2

(iii) (3, 5) → x = 3

(iv) (−4, −3) → x = −4

Question 6

Write the equation of three lines that are (i) parallel to the x-axis and (ii) parallel to the y-axis.

(i) Lines parallel to x-axis (form y = k):

y = 3

y = −4

y = −5/3

(ii) Lines parallel to y-axis (form x = k):

x = −9

x = 11

x = −8/11

Quick Reference — Equations and Their Graphs

y = b (horizontal line)

Parallel to x-axis Passes through (0, b) All points: (x, b) for any x

Example: y = 4 passes through (0, 4)

x = a (vertical line)

Parallel to y-axis Passes through (a, 0) All points: (a, y) for any y

Example: x = 3 passes through (3, 0)

y = 0

The x-axis itself Equation of the x-axis

Passes through origin, every point (x, 0)

x = 0

The y-axis itself Equation of the y-axis

Passes through origin, every point (0, y)

More Examples — y = 4 and x = −3 on the Cartesian Plane

Green line: y = 4 (horizontal, parallel to x-axis) · Red line: x = −3 (vertical, parallel to y-axis)

Notice the pattern: whenever the equation contains only x, the graph is a vertical line parallel to the y-axis. Whenever it contains only y, the graph is a horizontal line parallel to the x-axis. This simple rule is the key to solving all problems in Exercise 6.5 quickly and accurately.