Pair of Linear Equations in Two Variables — Class 10 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Pair of Linear Equations in Two Variables" — 9 important questions with detailed answers for CBSE board exam preparation.

✓ 100% Free ✓ No Login Needed ✓ NCERT / CBSE Aligned ✓ Download as PDF

TL;DR: Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Pair of Linear Equations in Two Variables" — 9 important questions with detailed a…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

🤖 Stuck on any question? Ask Syllab's free AI Tutor for a step-by-step explanation — instant, unlimited, no login.

Key Questions Covered:

Define a linear equation in two variables and state the general form.
A pair of linear equations is consistent if which condition is satisfied?
Solve the system of equations 2x + y = 7 and x - y = 2 using the elimination …
Solve the system 3x + 4y = 10 and 2x - 2y = 2 using substitution method.
Solve using the cross-multiplication method: x + 2y = 5 and 2x + 3y = 8
For what value of k does the system 2x - 3y = 7 and 4x - ky = 14 have infinit…
+ 3 more questions in the full chapter

Solutions Summary:

Question	Status
Define a linear equation in two variables and state the g…	✓ Solved
A pair of linear equations is consistent if which conditi…	✓ Solved
Solve the system of equations 2x + y = 7 and x - y = 2 us…	✓ Solved
Solve the system 3x + 4y = 10 and 2x - 2y = 2 using subst…	✓ Solved
Solve using the cross-multiplication method: x + 2y = 5 a…	✓ Solved
For what value of k does the system 2x - 3y = 7 and 4x - …	✓ Solved

Showing 6 of 9 questions

Q1: Define a linear equation in two variables and state the general form.

Linear Equation in Two Variables: A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, and c are real numbers and at least one of a or b is non-zero. General form: ax + by + c = 0 or ax + by = c where a = coefficient of x b = coefficient of y c = constant term Examples: - 2x + 3y - 6 = 0 - x - 2y = 5 - 3x + 4y = 12 Properties: - The graph of a linear equation in two variables is a straight line - The equation has infinitely many solutions (or...

Q2: A pair of linear equations is consistent if which condition is satisfied?

Consistency of a Pair of Linear Equations: Consider the pair: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 The pair of equations is CONSISTENT if it has at least one solution. Two cases of consistency: 1) Consistent and Independent: Condition: a₁/a₂ ≠ b₁/b₂ Number of solutions: Exactly one (unique) Geometrically: The lines intersect at one point 2) Consistent and Dependent: Condition: a₁/a₂ = b₁/b₂ = c₁/c₂ Number of solutions: Infinitely many Geometrically: The lines coincide...

Q3: Solve the system of equations 2x + y = 7 and x - y = 2 using the elimination method.

Given equations: 2x + y = 7 ... (1) x - y = 2 ... (2) Step 1: Add equations (1) and (2) to eliminate y (2x + y) + (x - y) = 7 + 2 3x = 9 x = 3 Step 2: Substitute x = 3 into equation (2) 3 - y = 2 y = 1 Step 3: Verification Substitute x = 3, y = 1 into equation (1): 2(3) + 1 = 6 + 1 = 7 ✓ Substitute x = 3, y = 1 into equation (2): 3 - 1 = 2 ✓ Therefore, the solution is x = 3, y = 1

Q4: Solve the system 3x + 4y = 10 and 2x - 2y = 2 using substitution method.

Given equations: 3x + 4y = 10 ... (1) 2x - 2y = 2 ... (2) Step 1: Simplify equation (2) by dividing by 2 x - y = 1 x = y + 1 ... (3) Step 2: Substitute x = y + 1 into equation (1) 3(y + 1) + 4y = 10 3y + 3 + 4y = 10 7y + 3 = 10 7y = 7 y = 1 Step 3: Find x using equation (3) x = 1 + 1 = 2 Step 4: Verification Substitute x = 2, y = 1 into equation (1): 3(2) + 4(1) = 6 + 4 = 10 ✓ Substitute x = 2, y = 1 into equation (2): 2(2) - 2(1) = 4 - 2 = 2 ✓ Therefore, the solution is x = 2, y = 1

Q5: Solve using the cross-multiplication method: x + 2y = 5 and 2x + 3y = 8

Given equations: x + 2y = 5 ... (1) 2x + 3y = 8 ... (2) Rearranging in the form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: x + 2y - 5 = 0 2x + 3y - 8 = 0 Here: a₁ = 1, b₁ = 2, c₁ = -5 a₂ = 2, b₂ = 3, c₂ = -8 Using cross-multiplication method: x/(b₁c₂ - b₂c₁) = y/(c₁a₂ - c₂a₁) = 1/(a₁b₂ - a₂b₁) Calculate each denominator: b₁c₂ - b₂c₁ = 2(-8) - 3(-5) = -16 + 15 = -1 c₁a₂ - c₂a₁ = (-5)(2) - (-8)(1) = -10 + 8 = -2 a₁b₂ - a₂b₁ = 1(3) - 2(2) = 3 - 4 = -1 So: x/(-1) = y/(-2) = 1/(-1) Ther...

Q6: For what value of k does the system 2x - 3y = 7 and 4x - ky = 14 have infinitely many solutions?

For a system of linear equations to have infinitely many solutions, the equations must be consistent and dependent. This means: a₁/a₂ = b₁/b₂ = c₁/c₂ For the given equations: 2x - 3y = 7 → a₁ = 2, b₁ = -3, c₁ = 7 4x - ky = 14 → a₂ = 4, b₂ = -k, c₂ = 14 Applying the condition: a₁/a₂ = b₁/b₂ = c₁/c₂ First, check c₁/c₂: 7/14 = 1/2 So we need a₁/a₂ = 1/2: 2/4 = 1/2 ✓ (This is satisfied) Now, we need b₁/b₂ = 1/2: (-3)/(-k) = 1/2 3/k = 1/2 k = 6 Verification: With k = 6: The second equatio...

Showing 6 of 9 questions. Visit the full page for complete solutions.