Chapter 3 — Pair of Linear Equations in Two Variables

Chapter 3 of the Class 10 Maths NCERT textbook, "Pair of Linear Equations in Two Variables", teaches students to solve such pairs using graphical, substitution, and elimination methods, and to classify a pair as consistent, inconsistent, or dependent from the ratios of its coefficients.

• What the solution looks like — Two linear equations correspond to two lines that may cross at one point, lie on top of each other, or run parallel. These three cases give a unique solution, infinitely many solutions, or no solution respectively.
• Three ways to solve — The chapter offers a graphical approach plus two algebraic ones: substitution replaces one variable using the other equation, while elimination scales equations so a variable cancels when added or subtracted.
• Predicting from coefficient ratios — Comparing the ratios of the coefficients tells you the outcome before solving — whether the pair is consistent with one solution, dependent with infinite solutions, or inconsistent with none.
• Modelling word problems — Everyday situations such as age puzzles, cost comparisons, and digit problems can be translated into a pair of equations, then solved with these methods to recover the unknown quantities.

1. 01A pair of linear equations can be consistent (has at least one solution) or inconsistent (has no solution); a dependent pair is always consistent with infinitely many solutions.
2. 02Graphically, two lines either intersect at one point (unique solution), coincide (infinite solutions), or are parallel (no solution).
3. 03The substitution method involves expressing one variable in terms of the other and substituting into the second equation to solve.
4. 04The elimination method multiplies equations by suitable constants to make coefficients of one variable equal, then adds or subtracts to eliminate that variable.
5. 05If a1/a2 ≠ b1/b2, the pair is consistent with a unique solution; if a1/a2 = b1/b2 = c1/c2, it is dependent (infinitely many solutions); if a1/a2 = b1/b2 ≠ c1/c2, it is inconsistent (no solution).
6. 06Real-life problems such as age puzzles, cost problems, and digit problems can be modelled and solved as pairs of linear equations.