Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables | NCERT Solutions & Board Questions (2025–26)

 Class 10 CBSE Maths – Chapter 3: Pair of Linear Equations in Two Variables

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🔹 Key Concepts and Formulas

✅ Linear Equation in Two Variables:

An equation of the form:
$ax + by + c = 0$
Where a, b, c are real numbers, and a and b are not both zero.

✅ Solution of a Pair of Linear Equations:

A solution is a pair (x, y) that satisfies both equations simultaneously.

✅ Graphical Method:

• Each equation represents a straight line.

• The point of intersection gives the solution.

Types of Solutions:

1. Consistent and Unique (Lines intersect at one point)

2. Consistent and Infinite Solutions (Lines coincide)

3. Inconsistent (Lines are parallel)

Condition Table:

Condition	Type of Lines	Number of Solutions
$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$	Intersecting lines	Unique solution
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$	Coincident lines	Infinitely many
$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$	Parallel lines	No solution

✅ Algebraic Methods:

1. Substitution Method

2. Elimination Method

3. Cross-Multiplication Method

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📘 NCERT Solved Examples (Highlights)

Example 1:

Solve graphically:
$x + y = 5 \text{ and } x - y = 1$
Solution: Solve for two points for each equation and plot. Intersection point is (3, 2).

Example 2:

Use elimination method:
$2x + 3y = 12 \text{ and } x - y = 1$
Multiply second equation by 3 and eliminate y.

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🔹 📄 Exercise 3.1 – Graphical Method

Q1. Solve and represent graphically:

$x + y = 5 \quad \text{and} \quad x - y = 1$

• Plot both equations.

• Intersect at (3, 2)
  ✅ Solution: $x = 3, y = 2$

Q2. Draw graphs and state solution type:

Check for parallel, intersecting, or coincident lines using condition table.

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🔹 📄 Exercise 3.2 – Algebraic Method: Substitution

Q1. Solve using substitution:

$x + y = 14 \quad \text{and} \quad x - y = 4$

• From first: $x = 14 - y$

• Substitute in second:
  $14 - y - y = 4 \Rightarrow y = 5$, then $x = 9$
  ✅ Solution: (9, 5)

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🔹 📄 Exercise 3.3 – Elimination Method

Q1. Solve:

$3x + 4y = 10 \quad \text{and} \quad 2x - 2y = 2$

• Multiply to make coefficients equal

• Add/Subtract equations to eliminate one variable

• Solve remaining linear equation
  ✅ Final Answer: $x = 2, y = 1$

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🔹 📄 Exercise 3.4 – Cross Multiplication Method

Q1. Solve:

$2x + 3y = 17 \quad \text{and} \quad 4x - y = 5$

• Apply cross multiplication:
  $\frac{x}{(b_1c_2 - b_2c_1)} = \frac{y}{(c_1a_2 - c_2a_1)} = \frac{1}{(a_1b_2 - a_2b_1)}$

• Plug values, solve to get x and y
  ✅ Solution: (x, y)

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🔹 📄 Exercise 3.5 – Applications (Word Problems)

Q1. The sum of the digits of a two-digit number is 9. If 27 is subtracted from the number, its digits are reversed. Find the number.

Let tens digit = x, unit digit = y

• Number = 10x + y

• Equation 1: $x + y = 9$

• Equation 2: $10x + y - 27 = 10y + x$

• Solve using substitution or elimination
  ✅ Answer: Number is 63

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🔹 Summary and Revision Notes

• Linear equations are straight-line equations in x and y

• Algebraic methods are more accurate than graphical method

• Always check type of solution using condition table

• Use substitution for easy expressions, elimination when coefficients match

• Word problems are modeled into linear systems and solved algebraically

Here’s a comprehensive list of past CBSE Board Exam Questions from Chapter 3: Pair of Linear Equations in Two Variables, arranged by marking scheme, updated for the 2025–26 exam pattern. These questions have appeared in actual board papers or official sample papers (2016–2024).

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📘 CBSE Past Year Board Questions – Class 10 Maths Chapter 3: Pair of Linear Equations

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✅ 1-MARK QUESTIONS

1. (2020) Write the condition for a pair of linear equations to be inconsistent.
   📌 Answer:
   $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

2. (2019) Find the value of k if the pair of equations has a unique solution:
   $2x + 3y = 7$
   $(k - 1)x + (k + 2)y = 3$
   📌 Answer: Use: $\frac{2}{k - 1} \neq \frac{3}{k + 2}$

3. (2021 – Sample) What type of pair of linear equations is represented by:
   $3x + 2y = 6$ and $6x + 4y = 12$?
   📌 Answer:
   $\frac{3}{6} = \frac{2}{4} = \frac{6}{12} \Rightarrow$ Infinitely many solutions

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✅ 2-MARK QUESTIONS

1. (2020) Solve the pair of equations:
   $5x - 3y = 11$,
   $-10x + 6y = -22$
   📌 Answer: Use elimination. Multiply first equation by 2.

2. (2017) If $x = 1, y = 1$ is a solution of the pair $2x + 3y = a$ and $4x + 6y = b$, find values of a and b.
   📌 Answer:
   Substituting:
   $a = 2 + 3 = 5$, $b = 4 + 6 = 10$

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✅ 3-MARK QUESTIONS

1. (2022) Solve by substitution:
   $x + y = 5$,
   $2x - 3y = 4$
   📌 Answer:
   From 1st: $x = 5 - y$, plug into 2nd:
   $2(5 - y) - 3y = 4$

2. (2020) Solve:
   $2x + 3y = 12$,
   $x - y = 1$
   📌 Answer: Use elimination. Multiply second equation by 3.

3. (2018) Find value of k for which the system has no solution:
   $kx + 3y = k - 3$,
   $12x + ky = k$
   📌 Answer:
   Use condition: $\frac{k}{12} = \frac{3}{k} \neq \frac{k - 3}{k}$

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✅ 4-MARK QUESTIONS

1. (2019) Solve:
   $3x - y = 3$,
   $9x - 3y = 9$
   📌 Answer:
   Check: $\frac{3}{9} = \frac{-1}{-3} = \frac{3}{9} \Rightarrow$ Infinite solutions

2. (2017) A boat goes 30 km downstream in 2 hours and 18 km upstream in 3 hours. Find speed of boat and current.
   📌 Answer:
   Let speed of boat = x, speed of stream = y
   Equations:
   $\frac{30}{x + y} = 2$,
   $\frac{18}{x - y} = 3$

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✅ 5-MARK WORD PROBLEMS

1. (2024 – Sample Paper) The sum of two numbers is 100. If one number is twice the other decreased by 10, find the numbers.
   📌 Answer:
   Let numbers be x and y
   $x + y = 100$,
   $x = 2y - 10$

2. (2020) A fraction becomes $\frac{1}{2}$ when 1 is subtracted from the numerator, and becomes $\frac{1}{3}$ when 1 is added to the denominator. Find the fraction.
   📌 Answer:
   Let fraction = $\frac{x}{y}$,
   Equations:
   $\frac{x - 1}{y} = \frac{1}{2}$,
   $\frac{x}{y + 1} = \frac{1}{3}$

3. (2016) The sum of the digits of a two-digit number is 9. If the digits are reversed, the new number is 27 less than the original. Find the number.
   📌 Answer:
   Let tens digit = x, units = y
   Equations:
   $x + y = 9$,
   $10x + y - 27 = 10y + x$

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🔁 Frequently Tested Topics:

• Graphical interpretation of solution types

• Solving by substitution, elimination

• Framing and solving word problems

• Conditions for unique/no/infinite solutions

• Forming equations from real-life contexts