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NCERT Solutions

Linear Equations in two Variables

Madhya Pradesh Board · Class 9 · Mathematics

NCERT Solutions for Linear Equations in two Variables — Madhya Pradesh Board Class 9 Mathematics.

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An infographic defining a linear equation in two variables, showing its general form (Ax + By + C = 0) and examples, with labels for variables and coefficients.
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6 Questions Solved · 2 Sections

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Exercise 4.1

1The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement. (Take the cost of a notebook to be ₹x and that of a pen to be ₹y).Show solution
Given: Cost of a notebook = ₹x, Cost of a pen = ₹y.

Statement: The cost of a notebook is twice the cost of a pen.

Forming the equation:
x=2yx = 2y

This can be written in standard form as:
x−2y=0x - 2y = 0

or equivalently 1⋅x+(−2)⋅y+0=01 \cdot x + (-2) \cdot y + 0 = 0.

Answer: The required linear equation in two variables is x=2yx = 2y (or x−2y=0x - 2y = 0).

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2Express the following linear equations in the form ax+by+c=0ax + by + c = 0 and indicate the values of aa, bb and cc in each case:
(i) 2x+3y=9.3‾52x + 3y = 9.\overline{3}5
(ii) x−y5−10=0x - \dfrac{y}{5} - 10 = 0
(iii) −2x+3y=6-2x + 3y = 6
(iv) x=3yx = 3y
(v) 2x=−5y2x = -5y
(vi) 3x+2=03x + 2 = 0
(vii) y−2=0y - 2 = 0
(viii) 5=2x5 = 2xShow solution
The standard form is ax+by+c=0ax + by + c = 0.

(i) 2x+3y=9.352x + 3y = 9.35

Rewriting: 2x+3y−9.35=02x + 3y - 9.35 = 0

Here, a=2,  b=3,  c=−9.35a = 2,\; b = 3,\; c = -9.35.

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(ii) x−y5−10=0x - \dfrac{y}{5} - 10 = 0

This is already in the required form: 1⋅x+(−15)y+(−10)=01 \cdot x + \left(-\dfrac{1}{5}\right)y + (-10) = 0

Here, a=1,  b=−15,  c=−10a = 1,\; b = -\dfrac{1}{5},\; c = -10.

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(iii) −2x+3y=6-2x + 3y = 6

Rewriting: −2x+3y−6=0-2x + 3y - 6 = 0

Here, a=−2,  b=3,  c=−6a = -2,\; b = 3,\; c = -6.

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(iv) x=3yx = 3y

Rewriting: x−3y=0x - 3y = 0, i.e., 1⋅x+(−3)y+0=01 \cdot x + (-3)y + 0 = 0

Here, a=1,  b=−3,  c=0a = 1,\; b = -3,\; c = 0.

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(v) 2x=−5y2x = -5y

Rewriting: 2x+5y=02x + 5y = 0, i.e., 2x+5y+0=02x + 5y + 0 = 0

Here, a=2,  b=5,  c=0a = 2,\; b = 5,\; c = 0.

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(vi) 3x+2=03x + 2 = 0

Rewriting: 3x+0⋅y+2=03x + 0 \cdot y + 2 = 0

Here, a=3,  b=0,  c=2a = 3,\; b = 0,\; c = 2.

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(vii) y−2=0y - 2 = 0

Rewriting: 0⋅x+1⋅y+(−2)=00 \cdot x + 1 \cdot y + (-2) = 0

Here, a=0,  b=1,  c=−2a = 0,\; b = 1,\; c = -2.

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(viii) 5=2x5 = 2x

Rewriting: 2x=5⇒2x−5=02x = 5 \Rightarrow 2x - 5 = 0, i.e., 2x+0⋅y+(−5)=02x + 0 \cdot y + (-5) = 0

Here, a=2,  b=0,  c=−5a = 2,\; b = 0,\; c = -5.

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Exercise 4.2

1Which one of the following options is true, and why? y=3x+5y = 3x + 5 has
(i) a unique solution,
(ii) only two solutions,
(iii) infinitely many solutionsShow solution
Correct option: (iii) infinitely many solutions.

Reason: The equation y=3x+5y = 3x + 5 is a linear equation in two variables xx and yy. For every real value we assign to xx, we get a corresponding real value of yy. Since there are infinitely many real numbers to substitute for xx, the equation has infinitely many solutions.

For example:
- x=0⇒y=5x = 0 \Rightarrow y = 5 → solution (0,5)(0, 5)
- x=1⇒y=8x = 1 \Rightarrow y = 8 → solution (1,8)(1, 8)
- x=−1⇒y=2x = -1 \Rightarrow y = 2 → solution (−1,2)(-1, 2)

Thus, a linear equation in two variables always has infinitely many solutions.

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2Write four solutions for each of the following equations:
(i) 2x+y=72x + y = 7
(ii) πx+y=9\pi x + y = 9
(iii) x=4yx = 4y
3Check which of the following are solutions of the equation x−2y=4x - 2y = 4 and which are not:
(i) (0,2)(0, 2)
(ii) (2,0)(2, 0)
(iii) (4,0)(4, 0)
(iv) (2, 42)(\sqrt{2},\ 4\sqrt{2})
(v) (1,1)(1, 1)
4Find the value of kk, if x=2, y=1x = 2,\ y = 1 is a solution of the equation 2x+3y=k2x + 3y = k.

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Frequently Asked Questions

What are the important topics in Linear Equations in two Variables for Madhya Pradesh Board Class 9 Mathematics?
Key topics in Linear Equations in two Variables include Linear Equations in Two Variables — Complete Concept Map, Complete Chapter Overview: Linear Equations in Two Variables, Linear Equations in Two Variables — Complete Concept Map. These are the concepts Madhya Pradesh Board Class 9 examiners draw on most — study them first, then practise related questions.
How to score full marks in Linear Equations in two Variables — Madhya Pradesh Board Class 9 Mathematics?
Understand the core concepts first, then work through the 32 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.
Where can I get free NCERT Solutions for Linear Equations in two Variables Class 9 Mathematics?
This page has free step-by-step NCERT Solutions for every exercise question in Linear Equations in two Variables (Madhya Pradesh Board Class 9 Mathematics) — written the way examiners award marks: given, formula, working, answer.

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