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Two Variables, One Line – Class 9 Mathematics notes, important questions and MCQs

NCERT Class 9 Mathematics, chapter 13. Notes, key terms, important questions with answers and MCQs.

Chapter overview

A linear equation in two variables, ax + by + c = 0, has infinitely many solutions, and all of them lie on one straight line.

Linear equations in two variables

Standard form ax + by + c = 0 with a and b not both zero.
A solution is a pair (x, y) that makes the equation true.
Every such equation has infinitely many solutions.

Graphs

Plot at least two (better three) solutions and join them: the graph is a straight line.
Every point on the line is a solution, and every solution lies on the line.
x = a is a line parallel to the y-axis; y = b is parallel to the x-axis; y = mx passes through the origin.

Key terms

Solution: an ordered pair satisfying the equation
Linear equation in two variables: ax + by + c = 0 with a, b not both 0
Graph of a linear equation: the straight line formed by all solutions (x, y) of ax + by + c = 0

Formulas

x = a: vertical line; y = b: horizontal line
Find intercepts: put x = 0 to get y, and y = 0 to get x

Important questions with answers

Check whether (2, 1) is a solution of 3x − 2y = 4.2 marks

3(2) − 2(1) = 6 − 2 = 4. Yes, (2, 1) is a solution.

Write four solutions of x + 2y = 6.3 marks

Choose x and find y = (6 − x)/2: (0, 3), (2, 2), (4, 1), (6, 0).

The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables, find two solutions, and describe its graph.5 marks

Let a notebook cost ₹x and a pen ₹y. Then x = 2y, i.e. x − 2y = 0. Solutions: (20, 10) and (30, 15). The graph is a straight line through the origin (0, 0) because the constant term is zero; every point on it is a solution.

Quick revision

Infinitely many solutions.
The graph is a straight line.
x = 0 is the y-axis; y = 0 is the x-axis.

Exam tips

Use intercepts (x = 0 and y = 0) for the quickest two points.
Check a third point to catch plotting mistakes.

Practice MCQs

How many solutions does 2x + 3y = 6 have?
1. one
2. two
3. none
4. infinitely many
Show answer
Answer: (d) infinitely many. Every point on the line is a solution.
Which is a solution of x + y = 5?
1. (3, 2)
2. (2, 2)
3. (5, 5)
4. (1, 3)
Show answer
Answer: (a) (3, 2). 3 + 2 = 5.
The graph of y = 3 is parallel to the
1. y-axis
2. line y = x
3. x-axis
4. none
Show answer
Answer: (c) x-axis. Horizontal line.
The line 2x + y = 4 meets the y-axis at
1. (2, 0)
2. (0, 4)
3. (0, 2)
4. (4, 0)
Show answer
Answer: (b) (0, 4). x = 0 ⇒ y = 4.
If (2, k) is a solution of 3x − y = 1, then k =
1. 5
2. 7
3. −5
4. 6
Show answer
Answer: (a) 5. 6 − k = 1 ⇒ k = 5.
Which line passes through the origin?
1. x + y = 1
2. y = 4
3. x = 3
4. y = 2x
Show answer
Answer: (d) y = 2x. (0, 0) satisfies y = 2x.
The taxi fare is ₹10 for the first km and ₹8 for each further km. For distance x and fare y, the equation is
1. y = 8x + 2
2. y = 10x
3. y = 8x + 10
4. y = 18x
Show answer
Answer: (a) y = 8x + 2. y = 10 + 8(x − 1) = 8x + 2.
The area of the triangle formed by x + y = 4 with the two axes is
1. 4
2. 8
3. 16
4. 2
Show answer
Answer: (b) 8. Intercepts 4 and 4: ½ × 4 × 4 = 8.

All chapters of Class 9 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.