System of Equations in JEE: Faster Than Cramer’s Rule? | All PYQs JEE Mains 2025

Main ideas / concepts

• The video discusses solving JEE Mains 2025 questions involving a system of three linear equations.
• The instructor states they will not use Cramer’s Rule. Instead, they use a geometry-based method based on the “family of planes.”
• Geometric insight (3 planes in 3D):
  • 3 planes intersect at a single point → unique solution
  • 3 planes intersect along a line → infinitely many solutions
  • No common intersection (e.g., parallel/consistent mismatch) → no solution

Method (family of planes) — step-by-step

1. Interpret the 3 given linear equations in variables (x, y, z) as 3 planes in 3D space.

2. Form a family of planes using any two of the three equations:

   • The instructor writes a parametric family by taking a linear combination of two equations.

   • Conceptual pattern:

     • If the equations are (E1) and (E2), construct a family such as: [ E1 + \lambda E2 = 0 ]

     • (or any equivalent parametric combination as used in the video) 3. Compare the resulting family-of-planes equation with the third plane. 4. Compute and compare ratios of corresponding coefficients:

       • Compare ratios for the coefficients of (x), (y), (z), and also the constant term. 5. Apply the decision rules (as stated in the video):
       • All corresponding coefficient ratios match, and the constant ratio also matches → the planes coincide in a way that produces infinitely many solutions (intersection along a line).

   • All corresponding coefficient ratios match, but the constant ratio does not match → inconsistent system → no solution.

   • If any coefficient ratio does not match → the intersection resolves to exactly one point → unique solution.

3. For each question’s target type (no solution / infinite solutions):

   • Set up the equalities needed to enforce that condition (e.g., match coefficient ratios but mismatch constants for no solution).
   • Solve for the parameter values (such as (\lambda), (m), (a), etc.).
   • Substitute back to find the requested variable(s).

Lessons conveyed across multiple problems

• The instructor repeatedly emphasizes:
  • The same concept applies across all these questions.
  • If you’re comfortable with the ratio-check logic from the family of planes, you can solve multiple PYQs quickly.
• They argue that the 3D plane interpretation is easier than determinant-based approaches like Cramer’s Rule.

What happens in the sample questions (high-level)

• Question 1 (target: no solution)

  • Construct family of planes from two equations and compare with the third.
  • Enforce “no solution”: coefficient ratios match but constants don’t.
  • Final statement (from subtitles): (\lambda = 17) and (m \ne 18).
  • Final option stated: Option 1.

• Question 2 (target: infinitely many solutions)

  • Build family of planes from two equations.
  • Use the infinite solutions condition via the required ratio relationships.
  • Final answer stated: Option 4 (with (m = 12) in subtitles).

• Question 3 (target: infinitely many solutions)

  • Again build a family plane and compare using coefficient/constant ratios.
  • Solve for parameter(s) (e.g., (t), (m)).
  • Final answer stated: Option 4 (subtitles indicate 57).

• Question 4 (target: infinitely many solutions)

  • Use ratio comparisons multiple times to derive relations (e.g., between (a) and (b)).
  • Conclude with an expression in requested variables and state:
    • Final numerical answer: 16
    • Option number: Option 1 (as written in subtitles)

Speakers / sources featured

• Single speaker/instructor (unnamed in the subtitles).