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Consistency of a pair of equations: Class 10 practice

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂: if a₁/a₂ ≠ b₁/b₂ the lines intersect (one solution); if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ they are parallel (no solution); if all three ratios are equal they coincide (infinitely many).. Read the method, follow the three worked examples, then try the six questions.

What you should know first

  • Fractions
  • Ratios

The idea behind consistency of a pair of equations

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂: if a₁/a₂ ≠ b₁/b₂ the lines intersect (one solution); if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ they are parallel (no solution); if all three ratios are equal they coincide (infinitely many).

Step-by-step method

  1. Write the three ratios.
  2. Compare a₁/a₂ with b₁/b₂.
  3. If equal, compare with c₁/c₂.
  4. State the type of solution.

Worked example 1 (Basic)

Classify: 1x + 1y = 2 and 5x + 4y = 9.

  1. a₁/a₂ = 1/5, b₁/b₂ = 1/4, c₁/c₂ = 2/9
  2. Unique solution (intersecting lines)
  3. Answer: Unique solution (intersecting lines)

Worked example 2 (Standard)

Classify: 4x + 2y = 5 and 16x + 8y = 21.

  1. a₁/a₂ = 1/4, b₁/b₂ = 1/4, c₁/c₂ = 5/21
  2. No solution (parallel lines)
  3. Answer: No solution (parallel lines)

Worked example 3 (Challenge)

Classify: 2x + 1y = 3 and 6x + 3y = 9.

  1. a₁/a₂ = 1/3, b₁/b₂ = 1/3, c₁/c₂ = 1/3
  2. Infinitely many solutions (coincident lines)
  3. Answer: Infinitely many solutions (coincident lines)

Common mistakes and misconceptions

  • Comparing only two ratios when the third decides
  • Calling coincident lines inconsistent

Practice questions

  1. (Basic) Classify: 1x + 5y = 6 and 3x + 10y = 13.
  2. (Basic) Classify: 4x + 2y = 7 and 17x + 8y = 29.
  3. (Standard) Classify: 3x + 3y = 1 and 6x + 6y = 3.
  4. (Standard) Classify: 1x + 3y = 7 and 3x + 9y = 22.
  5. (Challenge) Classify: 2x + 2y = 8 and 6x + 6y = 24.
  6. (Challenge) Classify: 1x + 2y = 6 and 2x + 4y = 12.

Answer key with full working

  • Q1: a₁/a₂ = 1/3, b₁/b₂ = 1/2, c₁/c₂ = 6/13 → Unique solution (intersecting lines) → Answer: Unique solution (intersecting lines)
  • Q2: a₁/a₂ = 4/17, b₁/b₂ = 1/4, c₁/c₂ = 7/29 → Unique solution (intersecting lines) → Answer: Unique solution (intersecting lines)
  • Q3: a₁/a₂ = 1/2, b₁/b₂ = 1/2, c₁/c₂ = 1/3 → No solution (parallel lines) → Answer: No solution (parallel lines)
  • Q4: a₁/a₂ = 1/3, b₁/b₂ = 1/3, c₁/c₂ = 7/22 → No solution (parallel lines) → Answer: No solution (parallel lines)
  • Q5: a₁/a₂ = 1/3, b₁/b₂ = 1/3, c₁/c₂ = 1/3 → Infinitely many solutions (coincident lines) → Answer: Infinitely many solutions (coincident lines)
  • Q6: a₁/a₂ = 1/2, b₁/b₂ = 1/2, c₁/c₂ = 1/2 → Infinitely many solutions (coincident lines) → Answer: Infinitely many solutions (coincident lines)

Syllabus note

This skill is listed under "Pair of Linear Equations" for Class 10 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

Frequently asked questions

Which chapter is consistency of a pair of equations in?

"Pair of Linear Equations", Class 10 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

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